LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


OR 


Received 
Accession  No.  (9  7j~  J  /      .    Class  No. 


ROBINSON'S  MATHEMATICAL  SERIES. 


KEY 


PROGRESSIVE 


HIGHER  AEITHMETIC. 


FOE  TEACHEKS  AND  PRIVATE  LEARNERS. 


NEW  YORK: 
IVISON,  PHINNEY,  BLAKEMAN  &  CO. 

CHICAGO:  S.  C.  GRIGGS  &  Co. 
1866. 


R  O  B  I  N  S  O  N'S 


The  'most  COMPLETE,  most  PRACTICAL,  and  most  SCIENTIFIC  SEE  ES  of 
MATHEMATICAL  TEXT-BOOKS  ever  issued  in  this  country. 


I.    Robinson's  Progressive  Table  Book.  ..... 

II.    Robinson's  Progressive  Primary  Arithmetic,  - 

III.  Robinson's  Progressive  Intellectual  Arithmetic,  •  - 

IV.  Robinson's  Rudiments  of  Written  Arithmetic,      - 
V.    Hobiiison's  Progressive  Practical  Arithmetic,        - 

VI.    Robinson's  Key  to  Practical  Arithmetic.  ..... 

vii.    Robinson's  Progresses  Si^ii^r  arithmetic,    - 
vm.    Robinson's  Key  to  Higher  Arithmetic,     - 
IX.    Robinson's  New  .Elementary  Algebra,      - 
X.    Robinson's  Key  to  Elementary  Algebra,  - 
XI.    Robinson's  University  A  Igebra,   -       -       - 
XII.    Robinson's  Key  to  University  Algebra,    ..... 

XIII.  Robinson's  New   University  Algebra,        ..... 

XIV.  Robinson's  Key  to  ]Mew  University  Algebra,  -       ... 
AV  .    Robinson's  New  Geometry  and  Trigonometry,    - 

XVI      "rZoblnson's  Surveying  and  navigation,     ..... 

XVII.    Hobinson's  Analyt.  Geometry  and  Conic  Sections, 
XVIII.    Robinson's  Differen.  and  Int.  Calculus,  (in  preparation  ,)- 
XIX.    Robinson's  Elementary  Astronomy,   ...... 

XX     Kobinson'3  University  Astronomy.     ...... 

XXI.    Robinson's  Mathematical  Operations,        • 
XXIi.    Robinson's  Key  to   Geometry  and  Trigonometry,  Conic 
Sections  and  Analytical  Geometry,      -       -       -       . 

Entered,  according  to  Act  of  Congress,  in  the  year  1SfiA  w 
DANIEL   W.  FISH   &  J,  H.   FRENCH, 

an*  I  ngnin  i:i  the  year  l>63,  by 
DANIEL    W.    FISH.    A.M., 

In  the  Clerk's  Office  of  the  District  Court  ,>f  the  TnHetl  States  for  tbe  Na  them 
District  of  tlio  New  York. 


PROMINENT    CHARACTERISTICS 

OF 

ROBINSON'S  MATHEMATICAL   SKRIES, 


The  books  of  this  Series,  although  many  of  them  have 
so  recently  been  published,  have  been  recommended  and 
adopted  by  hundreds  of  the  most  critical  and  su^vssful 
teachers,  for  the  following  reasons  : 

•    1.  For  the  philosophical  and  scientific  arrangement  of 
the  subjects. 

2.  For  the  conciseness  of  the  rules  and  the  brevity  and 
accuracy  of  the  definitions. 

3.  For  the  rigid  and  logical,  yet  full  and  comprehen- 
sive Analysis. 

4.  For  the  new,  original,  and  improved  methods  of 
operations,  not  found  in  most  other  works  of  the  kind. 

5.  For  the  very  largo  mnitbe.r  and  variety  of  practical 
examples — practical,   because   adapted    to   the   ordinary 
transactions  of  business  life. 

6.  For  their  typographical  execution,  substantial  bind" 
ing,  and  general  attractiveness. 

7.  For  the  easy  gradation  and  progressiveness,  not 
only  in  the  several  books  that  compose  the  series,  but  in 
the  arrangement  and  treatment  of  the  subjects  of  each 
book. 

8.  For  their  adaptation  to  the  various  grades  of  schol- 
arship in  all  our  schools. 

9.  For  the  general  unity  of  their  plan,  and  the  clear- 
ness and  perspicuity  of  their  style. 

10.  For,  scientific  accuracy,  combined  with  practical 
utility,  throughout  the  whole. 


KEY. 

ADDITION. 

(67,  page  25.) 


Ex.  3  An*.  1982738. 

Ex.  5.  An*.  3189. 

Ex.  7.  An*.  415184. 

Ex.  13.  An*.  16977. 

Ex.  17.  An*.  6076510. 


Ex.  4.  Ans.  435058. 

Ex.  6.  An*.  289142. 

Ex.  12.  An*.  3001623. 

Ex.  16.  An*.  1881. 

Ex.  20.  An*.  21184000 


Ex.  22.  Ans.  $1924950. 

ADDING  TWO  OR  MORE  COLUMNS  AT  ONE  OPERATION. 

(68,  page  28.) 

Ex.  5.  Number  of  churches,  35887; 

"         "   persons  accommodated,        13847902; 

Value  of  church  property,  $85774659. 

Ex.  6.  Pounds  of  butter,  312625306; 

«        «   cheese,  105735893; 

"       "  wool,  52516961; 

Bushels  of  wheat,  100485844. 


SUBTEACTION. 

(75,  page  31.) 

Ex.  5.  Ans.  174333815.      Ex.  6.  Ans.  2361650877. 

Ex.  7.  An*.  86602389426.    Ex.  8.  Ans.  9000989311. 

(25-  31) 

[5] 


6  SIMPLE   NUMBERS. 

Ex.  10.  An*.  86  years.  Ex.  13.     Ans.  $44656513 

Ex.  15.  Ans.  2121108  square  miles;  316636286  population, 

Ex.  20.  Ans.  2657043. 


TWO  OB  MORE  SUBTRAHENDS. 

(76,  page  34.) 
Ex.  5.  4568          Ex.  6.  4756+575+1404-84=5555 

1320  1200 

275  750 

320  96 


2653  Ans.  3509  Ans. 

Ex.  7.  $15760  Ex.  8.  $75860 


2175  45640 

3794  25175 

4587  

$5045  An? 

$5204  Ans. 

Ex.  9.  20000  Ex.  10.  398470 


11000  157548 

7000  143429 


Ans.  2000  square  miles.  97493  Ans. 

Ex.  11.  $61307088      Ex.  12.  $5760+$3575=$9335 


52889800  2746 

234000  4632 


$8233288  Ans.  $1957  Ans. 

(31-34) 


Ex.  13.  643J66 


MULTIPLICATION.  7 

Ex.  14.  $8186793 


65038 
114624 

Ans.  463504 


Ex.  15.  $12722470 


7821556 

424497 

2355016 

Ans.  $2121401 


5700314 
904299 

$1582180  Am. 


MULTIPLICATION. 

(85,  page  38.) 


Ex.  7.  Ans.  43506216. 
Ex.  16.^.  24500. 
Ex.  20.  Ans.  $909000. 


Ex.  8.  Ans.  48288058. 
Ex.  19.  Ans.  $31647000 


POWERS    OF   NUMBERS 

(91,  page  39.) 

Ex.  1.  72X72=5184. 

Ex.2.  12X12X12X12X12=248832. 

Ex.  3.  25X25X25=15625. 

Ex.4.  7X7X7X7X7X7X7=823543. 

Ex.  5.  19X19X19X19=130321. 

Ex.  6.  3X3X3X3X3X3=729 

(34-40) 


8  SIMPLE   NUMBERS. 

Ex.7.   Ans.   95=59049;   113=1331;  182=g24; 

244140625;  7862=617796;  94^=689869781056;  1004 
=100000000;  173=4913;  2515=996250626251. 

Ex.8.  83=512;  15*=225;  and  512x225=115200,  Ans. 
Ex.  9.  252=625;  34=81;  and  625x81=50625,  Ans. 
Ex.  10.  73X200=68600;  44xll*=30976;  and  68600— 
30976=37624,  Ans. 

CONTRACTIONS   IN   MULTIPLICATION. 

(98,  page  42.) 

Ex.  1.  736X6X4=17664,  Ans. 
Ex.  2.  538X8X7=30128,  Ans. 
Ex.  3.  27865X7X3X4,  or  27865X7X12,=2340660,^/U 

Ex.4.  7856X4X4X3X3, or 7856X12X12= 
1131264,  Ans. 

Ex.  5.  $185X8X7=$10360,  Ans. 

Ex.  6.  17740872X8X12=1703123712  cubic  feet,  Ans. 

(99.)  - 
Ex.  3.  Ans.  50000  dollars.         Ex.  4.  Ans.  100000000000 

(100,  page  43.) 
Ex.  3.  Ans.  10350000.  Ex.  5.  Ans.  192128000 

(102,  page  41 ) 

Ex.  1.  5784  Ex.  2.  3785 

246  721 


34704  26495 

138816  79485 


1422864  Ans.  2728985  Ans. 

(40-44) 


MULTIPLICATION. 


Ex.  3.  472856 
54918 


4255704 

8511408 
25534224 


25968305808  Ans. 

Ex.  5.  573042 
24816 


4584336 

9168672 
13753008 

14220610272  Ans. 

Ex.  7.  43725652 
5187914 


393530868 
787061736 
306079564 
612159128 

218628260 

226847922169928  Ans. 

Ex.  9.  2703605 
4249784 


18925235 
132476645 
113551410 

227102820 

11489737271320  Ans. 


Ex.  4.  43785 
7153 


131355 
656775 
306495 


313194105 

Ex.  6.  78563721 
127369 


707073489 
2828293956 
2121220467 
78563721 

10006582580049  An*  A 

Ex.  8.  3578426785 
64532164 


14313707140 
57254828560 
114509657120 
17892133925 
229019314240 

230923624151612740  An* 

Ex.  10.  9462108 
16824 


75696864 

227090592 
151393728 

159190504992  Ans. 


(44,  45) 


10  SIMPLE    NUMBERS. 

EXAMPLES    COMBINING   THE   PRECEDING   RULES. 

(Page  45.) 

Ex.  1.   #28  X  175=44900;   $37X320=$11840  5   $4900+ 

$11840=$16740,  Ans. 
Ex.  2.  $1200— ($364+$275+$150+$187)=$224;  and 

$224X5=81120,  Ans. 

Ex.  3.  29+32=61;  61X17=1037  miles,  Ans. 
Ex.4.  834X127=84318;  $47X97=$4559;  and  $4318+ 

$4559=18877,  cost;  127+97=224;  $40x224=88960, 

sold  for  ;  $8960— $8877=$83,  profit,  Ans. 

Ex.5.  77+56=133;  675— 133  =  542,  multiplicand.  3X 
156=468;  214—28=186;  468  —  186=282,  multiplier. 
542X282=152844,  Ans. 

Ex.6.  37+50=87;  87X6=522;  98+522=620,  multipli- 
plicand.  64—50=14;  14x5=70;  70—10=60,  multi- 
plier. 620X60=37200,  Ans. 

Ex.  7.  14X25=350;  9x36  =  324;  350—324  +  4324= 
4350,  multiplicand.  280—112=168;  376  +  42  =  418; 
418X4=1672;  168+1672=1840,  multiplier.  4350X 
1840=8004000,  Ans. 

Ex.  8.  $2751X29967=  $82439217 
$5030x23905=$120242150 


$37802933  Ans. 

Ex.  9.  1449075X203=294162225  acres  cultivated; 
1922890880—294162225=1628728655  acres,  Ans 

Ex.  10.  $2258+$105=$2363,  valuation  per  farm; 

$2363x1449075=13424164225,  Ans. 
Ex.11.  24X55=50000;  73=343; 

50000—343=49657,  Ans. 

(45,  46) 


M  ULTIPLICATION.  i  1 

V 

Ex.12.  15«=3375;  32X25=288;  208^=43264;  9x24 
=144.  3375+43264=46639  ;  288+144=432  ;  46639 
—432=46207,  Ans. 

Ex.  13.  4+27+256+3125+46656=50068,  Ans. 
Ex.  14.  1200000X400=480000000  pounds,  Ans. 

Ex.  15.  $2450,  value  of  house  ; 

$2450X6—  $500=14200r    «      «  farm; 
$2450X2=  4900,     «      «  stock; 


s.  $21550,  total  value. 
Ex.16.  1500  X  $7=$10500  ;  800x*10=$8000;  700 
X$6=$4200;  $8000  +  $4200=$12200;  $12200— 
$1050C=$1700,  Ans. 

Ex.  17.  ($450+$780+$1250+$2275)X3=$14265,^t*. 
Ex.  18.  $115X35000=$4025000,  Ans. 

Ex.  19.  $485X2500  =$1212500 
$1450X10  ==  14500 
$1250X25  =  31250 


$1258250  Ans. 

Ex.  20.  1401944X$20=$28038880,  value  of  double  eagles; 

62990 X $10=      629900,     «     «  eagles; 
154555 X   $5=      772775,     «     «   half  eagles; 
22059  X  $3=        66177,    "     "   $3  pieces. 

Ans.  $29507732,  total  value. 


DIVISION. 

(Ullage  49.) 

Ex.  1.    Am.  78972.  Ex.  2.     Ans.  121562. 

Ex.  3.     Ans.  152329.  ^x.  4.     ^ns.  6086847. 

(46-49) 


12  SIMPLE   NUMBERS. 

Ex.  9.     Am.  7198.  Ex.  10.  Ans.  7071. 

Ex.  11.  Ans.  15607.  Ex.  12.  Ans.  483402f 2. 

Ex.  13.  Ans.  1253974? |.  Ex.  14.  Ans.  5479f| jf. 

Ex.  15.  Ans.  2084768|f ff .  Ex.  16.  Ans.  24781™. 

Ex.  17.  Ans.  5851fff.  Ex.  18.  Ans.  591862f{}f 

Ex  19.  Ans.  15395919^f fii.       Ex.  20.  Ans. 

Ex.  21.  $147675^365^»404f  J  |  .4ns. 

Ex.  22.  $30732518--556= $55274i||  Ans. 

Ex.  23.  $5572470-v-287=$19416^  Ans. 

Ex.  24.  $8186793--27977=4292^f  of 

ABBREVIATED   LONG  DIVISION. 

(112  page  51.) 

Ex.  1.  204)77112(378  Ans. 
159 
163 

Ex.2.  72)65664(912  Ans. 
8 
14 

Ex.  3.  209)7913576(37864  Ans. 
164 
180 
133 

83 

Ex.  4.  698)6636584(9508  Ans. 
354 
55 

Ex.  5.  8903)4024156(452  Ans. 

4625 

.  1780 

(49-51) 


DIVISION. 

Ex.  6.    6791)760592(112  Am. 
814 

1358 

Ex.  7.    25203)101443929(4025^^  Am. 
631 

12786 
1854 

Ex.  8.  269181)1246038849(4629  An*. 
169314 
78062 
242262 

Ex.  9.  56240)2318922(41if  |>f  §  Ant. 
6932 
13082 

Ex.  10.  17300)1454900(84^^%  An*. 
7090 
1700 


CONTRACTIONS  IN  DIVISION. 

(121    page  57.) 
Ex.1.    3(435  Ex.2.     7)4256 

5)145  8)608 


29  An*.  76  Am 

Ex. -3.    9)17856  Ex.4.    2)15288 


8)1984  3)7644 

248  Am.  7)2548 

364  Ana. 
(51-57)       « 


14  SIMPLE   NUMBERS. 

Ex.  5,     8)972552                             Ex.  6.  9)526050 

7)121569  7)58450 

3)17367  2)8350 

5789  Ans.  4175  Am 
Ex.  7.    7)612360 


5)87480 
3)17496 

5832  Ans. 
Ex.8.    3)553 

5)184 1 

Quotient,    36  -  -  4x3=12 

13,  remainder. 
Ex.9.    3)10183 

5)3394  -  1 


7)678  -  -  -  4X3=12 


Quotient,         96   -  6X^X3=90 

103,  remainder. 
Ex.  10.  2)10197 

3)5098  1 

4)1699 1X2=  2 


5)424  -  -  -3X3X2=18 
Quotient,       84  -  4x4x3x2=96 

^  117,  remainder. 

(57) 


DIVISION.  15 

Ex.  11.  3)29792 


8)9930  2 


6)1241 2X3=    6 

Quotient,      206  -  -  -5x8X3  =120 

128,  remainder 
Ex.  12.  4)73522 

6)18380  - 2 

7)3063 2X4=    8 

Quotient,      437  -  -  -  4x6x4=  96 

106,  remainder 
Ex.  13.  3)63844 

5)21281  1 

9)4256  ....    1X3=    3 
Quotient,      472  .  -  -  8x5X3=120 

124,  remainder. 
Ex.  14.  2)386639 

3)193319  1 

4)64439 2X2=    4 

5)16109  -  .--  3XBX2=  18 

6)3221   -  -  4X4X3X2=  96 

Quotient,     536    5X^X4X3X2=600 

719,  remainder 
(57) 


16  SIMPLE    NUMBERS. 

Ex.  15.  4)734514 


6)183628   2 

7)30604 4X4=  16 


Quotient,       4372  18,  remainder. 

Ex.  16.  9)636388 

9)70709  7 

9)7856 5X9=  45 

Quotient,         872 8x9X9=648 

700,  remainder, 
Ex.  17.  5)4619 


5)923 4 


5)184 3X5=  15 


Quotient,       36 4x5x5=100 

119,  remainder, 
Ex.  18.  3)116423 

7)38807   - -          2 


7)5543 6X3=        18 


8)791  - 6X7X3=    126 


9)98 7X7X7X3=1029 

Quotient,  10  -  -  -  8x8x7X7x3=9408 


10583,  remainder. 
(57) 


DIVISION.  17 

Ex.  19.  5)79500 


5)15900 
5)3180  . 
7)636 


7)90 6X5X5X5=  750 


Quotient,         12  -  -  -  -  6x7X5x5x5=5250 


6000,  remainder. 

(122,  page  58.) 
Ex.  2.  AM.  79-&V  Ex.  4.  Ans.  230TV$ft. 

(123.) 

Ex.  2.  Ans.  27f3§.  Ex.  6.  Ans.  8206|f$j|. 

Ex.  7.  ^ws.  3005. 

EXAMPLES  COMBINING  THE  PRECEDING  RULES. 

(Page  59.) 

Ex.1.  $4X25=$100;  $3x36=$108;  $100+$108= 
$208 ;  208—8=26,  Ans. 

Ex.  2.  $10x12=8192;  $13xl7=$221;  $192+$221= 
$413,  cost;  $18  X (12+17)  =  $522;  $522— $413= 
$109,  Ans. 

Ex.  3.  $2X300+$750=$1350,  value  of  produce t 
$3X120+  $90=  $450,     «       «    stock; 


$900--25=$36,  AM. 

(57-59) 


18  SIMPLE   NUMBERS. 


Ex.4.  450+(24— 12)  X  5=510; 


(90-^-6)+ (8  X 11)— 18=30 ; 
510-r-30=17,  Am. 


Ex.5.  648 x  (3^X23)319  =  5184;  2910-f-15=194 ;  5184 
—194=4990,  dividend;  4375-^-175=25 ;  25x42+ 
32=409 ;  2863  ~ 409  =  7,  divisor.  Hence,  4990-*- 
7=712f ,  Ans. 

Ex.  6.  42X34=1428;  107100-^-1428=75,  Ans. 

Ex.  7.  Reversing  the  fifth  operation,  12x24=288; 
reversing  the  fourth  operation,  288-7-6=48 ; 
reversing  the  third  operation,  48 +(28 — 16)=60; 
reversing  the  second  operation,  60 — (72+l)=10; 
reversing  the  first  operation,  10x45=450,  Ans. 

Ex.8.  $60— $42=$18;  $36x50=81800; 
1800-^18=100  months,  Ans. 

Ex.  9.  251104-j-472=532,  Ans. 

Ex.  10.  30422=9253764,  Ans. 

Ex.  11.  453x307+109=139180,  Ans. 

Ex.  12.  $4+$7=$ll ;  $1276--$11=116,  number  of  each 
kind;  116x2=232,  whole  number  purchased; 
$7— $4=$3;  $3XH6=$348,  difference  in  cost. 

Ex.  13.  $950+$7500=$8450; 

$13982686-=-$8450=1654,  and  a  remainder  i 

of  $6386. 


Ex.  14.  854x4300000-5-860=3870000  tons,  Ans. 

Ex.  15.  ^3191876~400=57979f flf.     Hence,  by  this  est* 
mate,  57979  persons  died. 

Ex.  16.  508464-f-10593=48,  Ans. 
(59,  60) 


PKOBLEMS.  19 

Jlx   11,  $7680-r-$64=120,  number  sold; 

$960-f-120=$8,  gained  per  head; 
$64 — $8=$56,  cost  per  head ; 
$9800-^456=175,  number  bought. 

Ex.18.  $95X6+$1200=$1770;  $1770-:-30=$59,  Am. 
Ex.  19.  36X16=576,  number  of  days'  work  required; 

576-r-24=24,  number  of  days  24  men  will  require. 

Ex.  20.  $1650H-275=$6,  cost  per  barrel; 

($9— $6)  X  186=$558,  gain. 

Ex.  21.  840-5-(5+10+15)=28,  of  each  kind;  hence,  28X 
3=84,  whole  number. 

Ex.  22.  $965— ($5X160)  =  $165;  165--3  =  55  tons,  un- 
sold; 160+55=215  tons  bought. 

Ex.  23.  $3825-s-$85=45,  number  sold; 

$7560-v-($85-f-$5)=84,  whole  number  of  horses; 

($7560+$945)— $3825=$4680,  to  be  realized  on 

the  remainder. 

Hence,  $4680-K84— 45)=$120,  Am. 
Ex.  24.  $22360+$1742=$24102,  total  cost; 

$15480--$18=860;  860x2=1720,  No.  acres; 

$22360~-1720=$13,  original  cost  per  acre. 

PROBLEMS  IN  SIMPLE  INTEGRAL  NUMBERS. 

(127,  page  62.) 

The  following  are  the  general?  solutions : 
Prob  1.     Add  the  several  numbers. 

Prob  2.     Subtract  the  sum  of  the  given  numbers  from  the 
sum  of  all. 

Prob.  3.     Add  the  parts. 

Prob.  4.     Subtract  the  sum  of  the  given  parts  from  the  whole 
(60-63) 


20  SIMPLE   NUMBERS. 

Prob.  5,     Subtract  the  less  from  the  greater. 

Prob.  6.     Subtract  the  difference  from  the  greater. 

Prob.  7.     Add  the  difference  to  the  less. 

Prob.  8.     Subtract  the  subtrahend  from  the  minuend. 

Prob.  9.     Subtract  the  remainder  from  the  minuend. 

Prob.  10.  Add  the  subtrahend  and  remainder. 

Prob.  11.  Multiply  the  numbers  together. 

Prob.  12.  Divide  the  product  by  the  given  factor. 

Prob.  13.  Divide  the  continued  product  by  the  product  of 
the  given  factors. 

.Prob.  14.  Multiply  the  factors  together  in  continued  multi- 
plication. 

Prob.  15.  Multiply  the  multiplicand  by  the  multiplier. 

Prob.  16.  Divide  the  product  by  the  multiplicand. 

Prob.  17.  Divide  the  product  by  the  multiplier. 

Prob.  18.  Divide  each  number  by  the  other. 

Prob.  19.  Divide  the  dividend  by  the  divisor. 

Prob.  20.  Multiply  the  divisor  and  quotient  together. 

Prob.  21.  Divide  the  dividend  by  the  quotient. 

Prob.  22.  Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder. 

Prob.  23.  Subtract  the  remainder  from  the  dividend,  and  di- 
vide the  result  by  the  quotient. 

Prob.  24.  Multiply  the  final  quotient  and  the  several  divisors 
together. 

Prob.  25.  Divide  the  first  dividend  by  the  continued  product: 
of  the  final  quotient  into  all  the  given  divisors. 

Prob.  26.  Divide  the  dividend  by  the  several  divisors  sue- 
cessively. 

Prob.  27.  Add  together  the  numbers  comprising  each  set, 
and  subtract  the  less  sum  from  the  greater. 
(63,  64) 


FACTORING.  21 

Piob  28.  Multiply  together  the  factors  comprising  each  set, 
and  add  the  several  products. 

Prob.  29.  Multiply  together  the  factors  comprising  each  set, 
and  then  add  the  products  and  given  numbers. 

Prob.  30.  Multiply  together  the  factors  comprising  each  set, 
and  subtract  the  less  product  from  the  greater. 

Prob.  31.  Add  the  product  of  the  given  set  or  sets  of  fac- 
tors and  the  given  number  or  numbers. 

Prob.  32.  Subtract  the  sum  of  the  products  of  the  set  or 
sets  of  factors  which  form  the  less  number  from 
the  sum  of  the  products  of  the  set  or  sets  of  fac- 
tors which  form  the  greater  number. 


PROPERTIES  OF  NUMBERS. 

FACTORING. 

(142,  page  71.) 
Ex.  1.  Ans.  2,  5,  5,  43.  Ex.  2.  Ans.  3,  5,  163. 

Ex.  3.  Ans.  2,  2,  3,  3,  5,  5,  7. 

Ex.  4.  Ans.  2,  2,  2,  2,  2,  2,  2,  2  ,2,  2,  3,  7. 
Ex,  5.  Ans.  2,  7,  13,  13.  Ex.  6.  2,  2,  2,  5,  5,  5. 

Ex.  7.  Ans.  5,  5,  5,  5,  5,  5,  5,  5. 

Ex.  8.  Ans.  3,  3,  3,  7,  11,  13,  37. 

(144,  page  74.) 

Ex.  2.     Am.  2,  3,  7,  17,  17,  29.     Ex.  3.    Ans.  13,  17,  31 
Ex,  4.     Ans.  17,  19,  29.  Ex.  5.     Ans.  2,  11,  19,  487. 

Ex.  6.     Ans.  7,  83,  103.  Ex.  7.     Ans.  97, 103. 

*f        Ex.  8.     Ans.  3,  5,  59,  139. 

Ex.  9.     Ans.  3,  5,  7,  47,  181. 
(64-74) 


22  PROPERTIES    OF    NUMBERS. 

Ex.  10.  Am.  2,  2,  2,  2,  2,  2,  41,  149. 

Ex.  11.  Ans.  7,  11,  37,  79. 

Ex.  12.  Ans.  2,  5,  13, 17,  37. 

Ex.  13.  Ans.  13,  17,  29. 

Ex.  14.  Ans.  2,  2,  2,  3,  17,  19,  23. 

Ex.  15.  Ans.  2,  3,  5,  7,  19,  179. 

(145,  page  76.) 

Ex.  1.  120=1X2X2X2X3X5 

1,      2,  4,  8  Combinations  of  1  and  2. 

3,      6,          12,  24  «  «  1, 2,  and  3. 

5,  10,          20,  40 )         „  "123  and  5 

15,  30<         60,         120  J  1,4*,M 

Ans.  1,  2,  3,  4,  5,  6,  8,  10,  12,  15,  20,  24,  30,  40,  60,  120 

Ex,  2.     84=1X2X2X3X7 

1,         2,         4  Combinations  of  1  and  2. 
3,         6,       12  «  «   1,2,  and  3. 

7,       14,       28 •}          u  u   -•    o  o  flnj  7 

21,       42,       84}  1,  Z,d,and7. 

.  4tw.  1,  2,  3,  4,  6,  7,  12,  14,  21,  28,  42,  84. 

Ex.  3.     100=1X2X2X5X5 

1,  2,  4  Combinations  of  1  and  2. 

5>  10>  20l          «  «   1  2  and  5 

25,          50,        100 }  1*3,  and 

^rcs.  1,  2,  4,  5,  10,  20,  25,  50, 100. 

Ex.  4.    420=1X2X2X3X5X7 

1,  2,  4  Combinations  of  1  and  2. 

3,  6,  12  «  «   1,2,  and  3. 

if;          $          6?}         "  "1,2, 3,  and  5. 

7,          14,          28) 

21>  42>  84  I         «  «   1   2  3   5  and  7 

35,          70,         140  f  *'  ^  6>  *> and  7' 

105,        210,        420  J 

.      1 1,  2,  3,  4,  5,  6,  7, 10, 12, 14, 15,  20,  21,  28, 
"  {30,  35,  42,  60,  70,  84, 105, 140,  210,  420. 

(74-76) 


GREATEST   COMMON    DIVISOR.  23 

Ex.  5.     1050=1X2X5X5X3X7 

1,  5,  25  Combinations  of  1  and  5. 

2,  10,  50  «  "    1,  5,  and  2. 

8>  16>  751          «  "152  and  3 

6,  30,  150  j  I,  0,  ^,  ai 

7,  35,          175 1 

14,          70,          350  I  K 

21*        105*          525  i  '    >    ;    ;  a      ° 

42^        210?,        1050  J 

( 1,  2,  3,  5,  6,  7,  10,  14,  15,  21,  25,  30,  35,  42, 
I  50,  70,  75,  105,  150,  175,  210,  350,  .525, 1050. 


GREATEST  COMMON   DIVISOR. 

(149,  page  77.) 

Ex.  2.     2X3=6,  An*.  Ex.  5.    6x7=42,  Am 

Ex.8.    3X3X7=63,  ^ras.         Ex.9.     91,  Ans. 
Ex.  11.     4X3X7=84,  Ans. 

(150,  page  81.) 

Ex.  4.  Ans.  11.  Ex.  5.  ^is.  1. 

Ex.  7.  -4ns.  17.  Ex.  8.  Ans.  337 

Ex.  10.  In  the  operation  under  this  rule,  the  quotient  figure 
may  always  be  so  taken  that  the  product  shall  be  either 
greater,  or  less,  than  the  dividend;  in  either  case,  the  new 
divisor  will  be  the  difference  between  the  dividend  and 
product.  It  will  always  be  found  advantageous  to  use  that 
quotient  figure  which  will  give  the  least  number  for  a  new 
divisor.  In  the  first  operation  below,  the  second  quotient 
figure  is  1,  and  the  next  divisor  is  413690 ;  in  the  second 
operation,  the  second  quotient  figure  is  2,  which  gives  178593" 
for  the  next  divisor,  and  abbreviates  the  subsequent  work 
(76-81) 


PROPERTIES   OF  NUMBERS. 


FIRST  OPERATION. 

L005973 

4 

4616175 
4023892 

592283 

1 

592283 

413690 
357186 

1 

2 
3 

413690 

178593 
169512 

56504 

54486 

6 

9081 

2018 

4 

8072 

2018 

2 

1009 

SECOND  OPERATION. 

4616175 
4023892 


1005973 
1184566 

4 

2 

178593 

8 

169512 

3 

9081 

6 

8072 

4 

Am.  1009 

2 

592283 
535779 


56504 

54486 

2018 

2018 


Ex.  13.  An*.  47. 

Ex.  15.  In  order  that  the  bins  may  be  equal,  the  number 
of  bushels  contained  in  one  bin  must  be  a  common  divisor  of 
the  two  quantities.  And  in  order  that  the  number  of  bins 
may  be  the  least  possible,  each  must  contain  the  greatest  com- 
mon divisor  of  the  two  quantities.  Ans.  91  bushels. 

Ex.  16.  The  pannels,  to  be  of  uniform  length,  must  be  a 
common  measure  or  divisor  of  the  three  sides ;  and  to  be  of 
the  greatest  possible  length,  they  must  be  the  greatest  com- 
mon divisor  of  the  three  sides.  Ans.  11  feet. 

Ex.  17.  The  price  to  be  paid  by  each  is  the  greatest  com- 
mon divisor  of  the  three  sums,  $620,  $1116,  aud  81488, 
which  is  $124.  Hence,  B  can  purchase  $620n-$124=5 ; 
C  can  purchase  $1116  H-  $124  =  9;  and  D  can  purchase 

$1488-:-$124=12. 

Ex.  18.  The  greatest  common  divisor  of  14599  feet  and 
10361  feet  is  13  feet,  the  length  of  1  joint  in  the  fence, 
(14599+10361)X2=49920  feet,  the  entire  length  of  the 

(81) 


LEAST   COMMON   MULTIPLE. 


25 


fence.      49920  H- 13  =  3840,   the  number  of  joints  in  the 
fence;  and  3840x7=26880,  the  number  of  rails,  Ans. 


LEAST   COMMON   MULTIPLE. 

(155,  page  83.) 

Ex.  1.  2X2X3X11X7X5=4620,  Ans. 

Ex.2.  7X3X3X2X2X11X5=13860,  Ans. 

Ex.  3.  5X3X2X2X2=120,  Ans. 

Ex.  4.  7X5X3X2X2X2X2=1680,  Ans. 

Ex.  5.  7X5X5X3=525,  Ans. 

Ex.  6.  19X3X7X2=798,  Ans. 

Ex.  7.  2X2X2X2X2X2X3X3X5=2880,  Ans. 


Ex.  1.       5,  3,  2 

2,  2,  3,  7,  5 


(156,  page  85.) 

15.. 18.. 21.. 24.. 35.. 36.. 42.. 50.. 60 

3..   7..  4..  7..  6..  7..  5..  2 


5X3X2X2X2X3X7X5=12600,  Ans. 


,  Ex.  2.        2,  2,  3 
2,3,5 


6. .8. .10. .15. .18. .20. .24 
2..   5..   5..   3..   5..   2 


Ex.  3,        3,  5,  2 

3,  5,  7,  2 


2X2X3X2X3X5=360, 


9. .15. .25. .35. .45. .100 
3..  5..   7..   3..   10 


3X5X2X3X5X7X2=6300,  Ans. 


Ex.  4.        3,  3,  2 
2,  2,  5,  3 


18.. 27.. 36.. 40 
3..  2. .20 


3X3X2X2X2X5X3=1080, 
(83-85) 


26                   P. 
Ex.  5.         3,    3 
2,13 

ROPEKTIES   0 
12.  .26.  .52 

2.  .13.  .26 

2X2X3X13=156,  Am. 


Ex.  6.         2,  2,  17 

8,9 


32.. 34.. 36 

8..  9 


2X2X17X8X9=4896,  Ans. 

NOTE. — When  numbers  are  prime  to  each  other,  as  8  and  9  in 
the  above  operation,  their  product  will  be  their  least  common 
multiple. 


Ex.  7.         2,  2,  3 
2,3,3 


8. .12. .18. .24. .27. .36 


2.. 


3..   2..   9..  3 


2X2X2X3X3X3=216,  Ans. 


Ex. 


2,11 
2,3,5 


22. .33. .44. .55. .66 


3..   2..   5..   3 


2X11X2X3X^=660,  Ans. 

NOTE. — The  first  three  numbers  in  Ex.  7  and  the  first  two  ia 
Ex.  8,  above,  are  factors  of  remaining  numbers  in  the  exam- 
ples respectively.  They  might,  therefore,  have  been  omitted  in 
the  operations. 

Ex.9. 


2,  2,  3 

64..  84. 

.96..  216 

2,  2,  2,  2 

16..  7. 

.  8..  18 

3,3,7 

7 

9 

2X2X2X2X2X2X3X3X3X7=12098,  Ans. 

Ex.  10.  The  number  of  rods  that  will  furnish  whole  days' 
work  to  each  one,  must  be  some  common  multiple  of  14,  25, 
8  and  20 ;  and  the  least  number  of  rods  that  will  furnish 

(85) 


CANCELLATION.  27 

whole  days'  work  to  each  of  the  men,  must  therefore  be  the 
least  common  multiple  of  14,  25,  8  and  20. 

Ans.  1400  rods. 

Ex.  11.  The  least  common  multiple  of  the  prices,  $4,  $21, 
$49,  and  $72,  which  is  $3528,  Ans. 

Ex.  12.  When  all  the  men  work  together,  they  will  dig 
4 -f-  8 +6=1 8  rods  per  day.  The  ditch  must  therefore  be 
the  least  common  multiple  of  4  rods,  8  rods,  6  rods,  and  18 
rods,  which  is  72  rods,  Ans. 

Ex.  13.  The  least  common  multiple  of  11  feet  and  15 
feet  is  15  X  11=165  feet,  the  distance  the  carriage  must 
move  to  bring  the  rivets  up  together.  Hence,  165x575= 
94875  feet,  the  entire  distance  traveled ;  and  94875  feet-4- 
5280=17  miles  5115  feet,  Ans. 


CANCELLATION. 
(159,  page  87.) 

K  f> 

Ex.  2. 


-=80,  Ans. 

Jfrv/  i  oiv  9f'V  #  V  3f 
Af  /\/Lfc>  /\/>  /\  y>  XN/' 

O 

Ex.  3 


-=32,  ^Irw. 


61 
Ex.4, 

61,  Ans. 


(86-88) 


28  PROPERTIES   OF   NUMBERS. 

71  11 

<^X190X^ 

-=14839, . 


Ex.5. 


Ex.6. 


Ex.7. 


Ex.9. 


=403,  Ans. 


9, 
16 


13 


13, 


Ex.  10.     84+56=140  cents. 
240 


32  cents, 


240,  Ans. 


~\  cost  of  2  yards  of  the 

Ex.  11.     75X2+90=240  cents,  [-first  kind,  and  1  yard 

j  of  the  second. 


11  yards  of  the  second  kind ; 
11X2=22     "          "     first        " 


Ex.12. 


=60  cents,  Am. 


(88) 


NOTATION   AND   NUMERATION.  29 


FRACTIONS. 

NOTATION  AND  NUMERATION. 

(169,  page  90.) 

Ex.  3.  Ans.  |f.  Ex.  4.  Ans.  7^. 

Ex.  5.  Ans.  |f§.  Ex.  6.  Ans.  £f|f 

Ex.  7.  Ans.  25H<y><>.  Ex.  8.  Ans.  T1/y>52. 

Ex.  9.  Ans.  TWzj°oW 

Ex.  10.  Four  ninths;  seven  twelfths;  seventeen  thirty 
eighths;  forty-five  one  hundredths;  seventy -two  three  hundred 
seventy-fifths;  forty-eight  one  thousand  ninths;  eighty-four 
seven  thousand  eight  hundred  sixty-thirds  ;  four  hundred  fifty- 
six  five  hundred  thirty-sevenths. 

Ex.11.  Twenty  fourths;  eighty-seven  thirtieths  ;  ninety- 
five  one  hundredths;  forty-eight  twelfths;  seventy-five  four 
hundred  thirty-sevenths ;  one  hundred  seventy-five  halves ; 
four  hundred  thirty-six  fiftieths ;  seven  hundred  sixty-six 
four  thousand  eight  hundred  seventy-ninths. 

Ex.  12.  Four  hundred  sixty-seven  nine  hundred  thirty- 
sixths;  five  hundred  thirty-six  two  hundred  forty-eighths  ;  ten 
thousand  seventy-fifths;  seventy-five  ten  thousandths;  five 
thousand  seven  three  thousand  sevenths. 

Ex.  13.  One  hundred  fifty  Jive  hundred  thirty-sevenths ; 
four  hundred  thirty-six  nine  hundred  seventy  seconds ;  thir- 
teen thousand  seven  hundred  eighty-five  forty-seven  thousand 
nine  hundred  fifty-sixths  ;  one  hundred  fifty  thousand  seven- 
ty-two  four  hundred  seventy-five  thousandths ;  one  hundred 
thousand  one  two  hundred  thousand  seconds. 
(90,  91) 


30  FRACTIONS. 

SEDUCTION. 
(177,  page  92.) 

Ex.6.     ffl=ft,Ans.  Ex.7.     if|-||,  Ans. 

Ex.8.     &&=&,  An*.         Ex.9.     f  §ff  =Hf  ,  ^- 
Ex.  13    m%=llA™'  Ex-  15«  ^-  im 


(178,  page  93.) 

Ex.2.     <y=14,.4fw.         Ex.7.     *£f  i=28f-i=28f,    Ans. 
Ex.  9.     8f  f  3=253T^,  ^TW. 
Ex.  12.  iH*f=lTiSAT= 


(179,  page  94.) 

Ex.  2.     375^3  o^o  o?  ^^  EX.  3.     473='  ^  *,  Ans. 

Ex.  4.     36=3 


(ISO,  page  95.) 

Ex.  3.     57f=  4^4,  ^Ins.  Ex.  5. 

Ex.  8.     l^=^7,  Ans.  Ex.  10.  43|=1-J 

Ex.  11. 


(181,  page  96.) 

Ex.1.     24--3=8;  Ex.2.  96-5-12=8; 

2  _  16       Ane  7  _  56       /<,„« 

3  -  24?  -^±'<'6'  TU  -  U<>?  -afWI 

Ex.3.     51-7-17=3;  Ex.4.  78-^-13=6; 

If  =|  f  ,  ^TIS.  A=W. 

Ex.  5.     3000--375—8  ;  Ex.  6.  8-*-4=2  ; 

3%-3409060.   ^'  7|  =  V  = 

Ex.7.     162\=16i=y; 

176-^-4=44;   \5^2T8^°, 

Ex.8.     363--llr=33; 


Ex.9,     42V7---  ^6;  ^=*  ^  =*  %**  ,  Ans. 
(92-96) 


REDUCTION.  81 

page  96.) 


, 

Ex.2.     i,f=Jf,  If,  An*. 
Ex.6,     i,  >,  &=&,  A3T?  3Vn  ^ 

(184,  page  98.) 
Ex.  1.     2X2X2X^=40,  least  common  denominator; 

f  ?  A=Jfc  l§?  ^*- 
Ex.  2.     3x2X2=12,  least  common  denominator; 

I.  f  >  t=A.  A»  H>  Ans- 

Ex.  3.     5x3X2X2=60,  least  common  denominator, 

t?  T72>  Ti=ife  l§>  f  &  ^w*« 
Ex.  4.     2  X  2  X  2  X  3  X  3=72,  least  common  denominator ; 

2     g     3 48      64     27 

^?   V)   8 72?   72^   75^ 

Ex.5      A=|;^==A. 


7,2 
2,3 


7. .12. .42 
6..   3 


7X2X2X^=84,  least  common  denominator. 

3       5        13  _  36      35     26       An* 

T   12?    42  -  84>   84?  54?  ^nS' 

Ex.  6.     13X3X2=78,  least  common  denominator; 


2       4       25       4     _  52     24     75       8 
3>   T3?  2^'    3~5>  -  7H»    78"?  T8?   78"? 


Ex.7.     5  X  3  X  2  X  2  X  2=120,  least  common  denominator  ; 

V?  T75>  A?  |J=*4§,  i5^?  T¥TF?  AV  ^- 
Ex.  8.     ||=i|;  7X2X2X2X3=168,  least  com.  denom.,- 

20   9   17  _  160   27   102 

27?  ^6?  28"  -  T5f  ?  T68"?  T5F? 

Ex.9.    f§=|;  /A=A;  il=A; 


2,2,3 

2,2,2 


8.. 24.. 32 
2..   2..   8 


3X25=96,  least  common  denominator; 

0 
6?  " 


§5         7  _  60     20     21 
>  23?   32  —  3       " 


(96—98) 


32  FRACTIONS, 

EX.  10.  £V=ft;  iW=tt;  iiS=^i; 

13x17=221,  least  common  denominator; 


A,  K,  262°T=3¥T>  ill, 


EX.  11.  ^=3*,;  ff?=II;  tttf=H; 

53X23=1219,  least  common  denominator; 


Ex.  12.  7X^X2—70,  least  common  denominator; 


V9. 


Ex.13.  T9^-ilx3!l;  f4H=fl;  fHi=«5«=i 

97X59X31X2=354826,  least  com.  denom.; 
T983^  fi  Ill-lf  Ilil,  4n 

Ex.  14.  23X33X5X7=7560,  least  com.  denom.; 

7?   T5^  T^>  2TJ  "35^?   4  O1^ 

5400     jg930     1008     2240      1944      3213      Avt* 
7^560?    7^560?  7560?    7560?    7560?  ^Se'O*  -a71*' 

Ex.  15.  13XTX2X2=364,  least  com.  denom.; 

435715  _  208       84         65         49         30 
7?   T3?  28?    55?   T82  -  3^4?   3^4^    354?    354"?   304? 

Ex.16.  ^=A;  11=4; 

5X3X7=105,  least  com.  denom.; 


ttfc 


ADDITION. 

(186,  page  100.) 

Ex.  1. 
Ex.2. 
Ex.  3.  ^^iJLLL»=4fi==2§,  Am. 

Ex.  4.     7+8+2+5+4         =26 


28|, 
(98-100) 


ADDITION.  33 

Ex.  5.     37+12+13  =62 


64|,  Ans. 
Ex.  6. 

EX.  7. 

Ex.  8.     i+f+T>s=u±4JJ>±3=ff,  _ 

EX.  9.       TS3+If  +57,=25±f|±21=          , 

Ex.  10.  ^+|+i^+||=28±«35t-a±^==2gY=,2f|,  Am. 

Ex.  11.  3+^+12+11+14= 

189+196+1^4+207+309—1^^6=4^,  Ans. 
Ex.  12.  3+4+2=  9 


Ex.  13.  16+24=40 


Ex.  14.  1+2+3+4+5=15 


1844, 
Ex.  15.  4+8+2      =14 


Ex.  16. 
Ex.  17. 

Ex.18 

.9,9  +  44  +  -  ,          > 

Ex.  19.  4+^+A+|= 

|««+«  ^=\^.  ^m^,  Ans 
(100) 


34  FBACTIONS. 

Ex.  20.  41+105+300+241+472=1159 

i  +  §  +  I  +  I  +  i  =     2! 


1161f 


Ex.  21.  4+2+  1  +  2  +  5  +7+4+6=31 


Ex.  22.  36+42+39+51=168 


169||  pounds, 
Ex.  23.  4+  3  =7 


Ex.  24.  ^+^=11-^=4 

Ex.  25.  ^+  1+1+^=^=1  of  a  dollar, 

Ex.  26.  46+64+76=186 


187||  yards; 

$127+$226+$312=$ 


$666{|,  received  for  the  whole. 


SUBTRACTION. 


(188,  page  102.) 

Ex.1.     T?f = ft,  An,.  Ex.2.     ^p=^=^  Am. 

Ex,  3.     JLf^L=ff=4,  Ans. 

(100—102) 


SUBTRACTION.  35 


rv    A 
UX.  4. 

Ex.  5. 
Ex.  6. 

EV    7          9          Ifi  _  81—32  _  49  _  7 
&X-  «•      T3  —  B3  --  T2g-  —  T25  —  T5' 

V-a-    8          14        1  9  _  7  0—  S  7  _   13  _  1 
JliX.  0.       -33  -  gs  -  -T95  --  T¥5  -  Tg, 

Ex.  9. 

Ex.  10.  ^-^^i^U-^f^il,  Ans. 

Ex.  11. 

Ex,  12. 


Ex.  13.  16|  Ex.  14.  36^ 


* 


Ex.  15.  25T^=25|J  Ex.  16.  75 

14i|=14|§  4f 

10§$=10|,  Ans.  704, 

Ex.  17.  18f=18T4g  Ex.  18.  2637?=26T33% 


12T7F,  ^4«s.  $fo=U,  An* 

Ex.  19.  2841=28,% 


Ex.  20.  78^=78^ 
32|  =32jf 

45{|=45|,  Am. 

Ex  21.  364—  7-l'3=19&,  ^»w. 
Ex  22.  97|—  184=79||  ,  Ans. 

Ex  23.  126^+240|=366||; 

560|-366|f  =198|J,  Am 
(102) 


Kf>  FRACTIONS. 

Bx.  24,  -g  +  T^  +  ig—  7';?  >  II     •?!—  f!> 

Ex.  25.  |—  f  =f  f  ,  ^Lw«. 

Ex.  26.  81£—  14g=16f  gallons,  ^Irw. 

Ex.  27.  $140|  $  775  1 

456 


$597^  >  bought  for;   $1291ff,  sold  for; 
$1291f  f—  6597^= 


MULTIPLICATION. 

(193,  page  105.) 

Ex.  1.     |xf=!=2f  ,  Am. 
Ex.2. 

Ex.  3. 

%  5.     YXA=15;-  fX'A=l=2i5  if^X  1=630  j 


Ex.6.  J}X8= 

1?v     7  lls/20  -  4.     16\v51  -  S 

nx.   I.  75X33  —  -g,   T7AT5  —  5- 

V  v     8  7   s/  I  7  _  17. 

^IA.  O.  75^42  —  72  > 


Ex.9.     i,8x\5=10,  ^ms.       Ex.  10.  f  xffi=|=2J, 

Ex.  11.  1RT 

Ex.  12.  | 

Ex.  13. 

Ex.14.  ii 

Ex.  15.  ^ 

Ex.  16.  T\  X  fT  X  V  X  V  X  fj,=  tf  =lg, 

Ex.  17.  ' 

Ex.18. 


(102-106) 


MULTIPLICATION.  37 

Ex.19.  7f—  2f=55fcj  |+J=1T2F; 


Ex.  20.  »xf  XlHb  -4«W       Ex.  21.  | 

Ex.22.  | 

Ex.23. 

Ex.  24.  iXf  Xf  Xf  Xf  Xf  XiXf  XT9t5=T\» 

Ex.25.  W=f?&l;  WI=5«5?!J  l«l 

llxMXfl^ff  xmtti=fti>  Ans- 
Ex.26.  i|5|iXf0if 

EX.  27.  4^!llx!^i 

Ex.  28.  $3|  X7=$25|, 

Ex.  29. 

Ex.  30.  4XlOX6=25f,  Am. 

Ex.  31.  10^x^x^X^111^=2134^1 

Ex.32.  $9fX£X|=44,  Ans. 

Ex.  33.  «T9gXlf=4T%>  ^ws-         Ex.  34.  «| 

Ex.  35.  156f  Xf  XiX!=47,  Ans. 

Ex.36. 


Ex.  37.  l|X6x8T9o=94i|,  Ans. 

Ex.  38.  fXl21f=104f  ;  |x48f=36i; 

104f  +36^—  75=65T%  ; 

150^—  65T9ij=84|,  multiplicand; 


X  3=4^,  multiplier; 


Ex.  39.  JxiXf==§ 

Ex.  40.  |—  (|X|)=254,  A's  stare; 

I  X  I—  (I  X|  X  |)=T5g>  B'8  share  J 
^Xf  XS—  (!X!X!X|)=455,  C's  share; 
|X|X|X|=254,  D's  share. 
(106,  107) 


38  FRACTIONS. 

Ex.41.  2iXl=A;  |X41X(|)2-15;  (3f)3-(3f)3 

=8511; 


DIVISION. 
(195,  page  108.) 

Ex.2     j?Xi=T2T5  ilfX^gfs- 
Ex.  3.     iT<>XI=35,  .4ns. 

Ex.4.     YX|=1|a=37i;  fxV8=V=7i- 

Ex.5.     VX-A=36i^«»-  Ex-6-    if  Xf=|, 

Ex.7.    i$Xf=fj  HXY=fi=l{f;  VXA=I- 

Ex.8. 

Ex.9. 

Ex.  11.  £XT5TX  V  X  ^=1,  Am. 

EX.  12.  ^xAx|x^=V!/.==i 

Ex.  13.  iX356XT3(5Xil=|f=lT7T,  An*. 

Ex.  14.  VXJXT\X4=$>  ^ws- 

Ex.  15.  L3  x  V9  X  f  X  35  X  T83=25; 

Ex.  16.  T63xHXfXf 

Ex.  17.  MIX  W^il 

Ex.  18.  VxWxxBVX|f 

Ex.  19. 


Ex.22.  -  =|XMX1T8XT2T=2,  Am. 


lOf 
Ex.  23.  7+3|=10f  ;  -  ^8H5Xj|=.2S==7i,  Am. 


4  ^     17 


(107-110) 


DIVISION.  39 

1_ 
Ex.  25.    $—  §=&      -1-orrjVXf  Xj=i  ^«- 

IXI 

fxf 

Ex.  26.  6£-5T<s=if  ;  -  -=f  Xf  X|f=M,  AM. 
If 


Ex.  27. 

Ex.  28.  16*41=X1=^=171,  Am. 

Ex.  29.  $30-s-$=$V)Xf=$Jifi= 

Ex.  30.  *A-*-(iX»i)=AXf  X|=^=2|  pints, 

Ex.  31.  $2J—TV= 

Ex.  32. 

Ex.33. 


Ex.34.  26^-r-|=i|ix|=638= 

Ex.  35.  27-^2!=YxA=23433 

Ex.  36.  148|-!-16|i==i|Ax^=^=±«H;  Am. 

Ex.  37.  IXfXf  Xi=|«=Ui,  An,. 
Ex.38.  28—  7|=20|;  20^x|=12i§;  720—  12{|= 
707T3g5  A-s-«=H;  40i  +  li=41|j  41|X 

(i)4=2§l  > 
271T|T, 

Ex.  39. 


Ex.  40  iXT^Xyxf  X347X3'VX^XT8TXf  X|XA 


(110) 


40       .  FRACTIONS. 


GREATEST   COMMON   DIVISOR   OP   FRACTIONS. 

(198,  page  112.) 

Ex.  1.     The  greatest  common  divisor  of  7,  14,  and  28  is  7 
the  least  common  multiple  of  9,  27,  and  45  is  135  ; 

Ans.  Tf  5. 

Ex.2.     3*,l$,f!=V>VM*; 

the  greatest  common  divisor  of  16,  12,  and  24  is  4  ; 

the  least  common  multiple  of  5,  7,  and  35  is  35  ; 

Ans.  sV 
Ex.3.    4,2f,2f,^=f,^i_2?_2_. 

greatest  common  divisor  of  4,  20,  12,  and  2  is  2  ; 

least  common  multiple  of  1,  9,  5,  and  45  is  45  ; 

Ans.  ft. 
Ex.  4.     109£,  1224=&ffi.,  £|p; 

greatest  common  divisor  of  546  and  858  is  78  ; 

least  common  multiple  of  5  and  7  is  35  ; 

if  =2^,  Ans. 
Ex.  5.     The  measure  will  be  the  greatest  common  divisor  of 

18|  feet  and  57^  feet,  which  is  2-^  feet,  Ans. 
Ex.  &     The  greatest  common  divisor  of  134|  gallons,  128  1 

gallons,  and  115|  gallons,  is  6T52  gallons,  the  capac- 

ity of  the  casks  required.     Hence, 

134|-j-6T52^=21,  number  of  casks  for  the  first  kind  $ 

128i-r-6T55=20;        «  «         «     second     « 

115i-s-6/3=18,        «  «        «       third     « 


59, 


(112) 


PROMISCUOUS  EXAMPLES.  41 


LEAST   COMMON    MULTIPLE   OF   FRACTIONS. 

(2O1,  page  113.) 

Ex.  1.     The  least  common  multiple  of  2,  7,  14,  and  8  is  56  ; 
the  greatest  common  divisor  of  5,  10,  15,  and  25  is  5  ; 


Ex.  2.  The  least  common  multiple  of  7,  35,  and  49  is  245  ; 
the  greatest  common  divisor  of  24,  36,  and  60  is  12  ; 

\V=20T^,  Ans. 

Ex.  3.     2||,  Ift,  $h=&,  y£,  rfft  ; 

least  common  multiple  of  72,  112,  and  63  is  1008  ; 

greatest  common  divisor  of  25,  75  and  100  is  25  ; 

^F=40^,  Ans. 
Ex.  4.     Least  common  multiple  of  1,  2,  3,  4,  5,  6,  7,  8,  9,  is 

2520  ;   greatest  common  divisor  of  2,  3,  4,  5,  6,  7, 

8,  9,  10,  is  1.     Ans.  2520. 

Ex.  5.  The  train  must  move  a  distance  equal  to  the  least 
common  multiple  of  15T5g  feet  and  9|  feet,  which 
is  459|  feet,  Ans. 


PROMISCUOUS  EXAMPLES 
(Page  114.) 

Ex.  1.     ?Xf=f  ;  135-9=15;  |=T%,  Ans. 

Ex.  2.     48--4^12;  48-j-6=r8  ;  48—8=6;  48—12=4; 


Ex.  3.     11,  f  ,  2,  ^,  |  of  |,  |  of  i=|,  f  ,  f  ,  /o 
3X5X3X2X7=630,  Ans. 
(113,  114) 


42  FRACTIONS. 


Ex.  8.  I — f— II-  A  number  diminished  by  }|  of  itself 
will  leave  a  remainder  of  1 — ||=|f  of  itself;  hence, 
141-5-|f=283^  Am. 

Ex.  9.     ^-f-f+^jo s .  i_ifa=Tr7_  of  his  money  left . 

hence,  8119-4-^^=8840,  Am. 

Ex.  10.  $42X,VX^X887-I311,  Am. 

Ex.  11.  Since  the  less  is  f  of  the  greater,  their  difference  is 
Jf  —  f =2  of  the  greater ;  hence,  25-&  -i-  f  —  SU-^, 
the  greater  number;  89T\— 25T7g=63|,  the  less. 

Ex.  12.  The  two  shares  together  must  be  |-|-|=y  times  the 
greater  share;  hence,  12000-r-1/— $1125,  the  great- 
er share;  $2000— $1125=1875,  the  less  share. 

Ex  13.  fx|X|= 10,  Am. 
Ex.  14. 
Ex.15. 

Ex.  16.   y  X  I— bushels  of  corn  that  can  be  bought  for  $15 ; 

V5XIX|X|=18  bushels  of  barley,  Ans. 

(114,  115) 


PROMISCUOUS   EXAMPLES.  43 

Ex.17. 


Ex.  18.   y  X|=  the  number  of  yards  of  cloth  1  yard  wide; 
VX|X|=  34£  yards,  Ans. 


Ex  19    f  XS=2  of  the  foundery  sold  for  $2570f  ; 
henee,  $2570f  X2=$5141£,  Ans. 

Ex.  20,  $10000x|XfXV2Xf  XIXV^^OOO,  vessel; 
$10000+$12000:=$22000,  Ans. 

Ex.  21.  The  second  son  had  |  of  f  =T63  ;  the  third  son  had 
1  —  (f  +T62)—  554  >  tne  difference  between  the  shares 
of  the  first  and  second  is  T52  —  f  =2*4  ;  hence,  $500 
_^i_*=$1200<T,  whole  estate;  $12000x^=82500, 
third  son's  share. 

Ex.  22.  $6T°XTV=$8,  cost  of  1  ton;  $78-M&8=9f,  Ans. 

Ex.  23.  16^30=  VXA=ii;  A™> 

Ex.  24.  I  of  the  whole-{-12|  acres=lst  and  2d  sons'  shares; 
3  a     ((       u    _^_i2i    «    ^3d  son's  share  ; 
I  of  the  whole  -j-  24^  "    =the   whole  ;     therefore, 
24£  acres  must  be  |  of  the  whole;  24.  JX  4=98  acres, 
the  whole;  and  98Xf+12|=49  acres,  Ans. 

Ex.  25.  $|X|XJX|=*|,  Am. 

Ex.  26.  $3500—  $>740=:$2760,  his  money  before  gaining; 

hence,  $2760-r-|=$4600,  money  invested  ;  and 

$4600  X  §=$1840,  lost,  .Ans. 
Ex.  27.  |  of  f=4,  Ans. 

Ex.  28.  Since  A  can  do  |  of  the  work  in  1  day,  and  B  can 

do  1  of  it  in  1  day,  they  can  both  do  '-f-i^^  of 

the  work  in  1  day;  and  if  ^  be  done  in  1  day,  ^, 

or  the  whole  work,  will  require  2T4=z3|  days,  Au*. 

£jOTEp  —  The  time  required  to  perform  any  piece  of  wo  k  will 

always  be  the  reciprocal  of  that   iractioii  of  the  work  pu  jormed 

in  1  unit  of  time. 

(115,116) 


44  FRACTIONS. 

Ex.  29.   %5Xf  Xi=13  barrels,  sold  at  $4  per  barrel; 
13+5=18  barrels,  Am. 

Ex.  30.  The  number  will  be  the  least  common  multiple  of 
f,  f ,  f  and  |,  which  is  60,  Ans. 

Ex.  31.  According  to  the  note  above,  A  will  travel  round 
the  island,  and  be  again  at  the  point  of  starting; 
once  in  |  days,  B  once  in  ^  days,  and  C  once  in 
.«y  days ;  and  the  least  common  multiple  of  |  days, 
L7  days,  and.  y  days,  is  -§-|^=178^  days,  Ans. 

Ex.  32.  The  sum  of  the  distances  traveled  by  the  two  men 
is  64J  miles,  and  the  difference  of  theee  distances  is 
5^  miles.  Hence,  by  Problem  33,  page  64,  of  the 
Arithmetic,  we  have 

64f  +  5!=76;Jj  70^-5-2=351  miles,  the  greater 
journey;  64|— 5^=59| ;  59£-*-2=2'9f  miles,  the 
less  journey. 

Ex.33.  l^+|=li;  li-*-2=f,  the  greater; 
Ilk— 1=&>  T7^2=^,  the  less. 

Ex.  34.  1— TVe— {Jl,  the  swm  of  B's  and  O's  shares.     And 
since  77^  is  the  difference  of  B's  and  C's  shares,  we 
^ve  iil+T7H-=f ;  *-s-2=A,  B^s  share; 
iH-A=if;  if^-2=J|,  C's  share. 

Ex.  35.  The  reciprocal  of  |  is  | ; 

reversing  the  fourth  operation,  |  X  26^=i%  > 
reversing  the  third  operation,  T3Q-j-|=y7^  ; 
reversing  the  second  operation,  T7a — J=T8; ; 
reversing  the  first  operation,  T8^x$yf =$|,  Ans. 

Ex  36.  ?!  is  a  quotient  and  If  a  divisor,  and  ]36xf=  y\6, 
the  dividend ;   \\5  is  a  product  and  5|  a  multiplier, 
and    j345-:-257  — ff>  khe  multiplicand;  f|  is  a  re- 
mainder and  |  a  subtrahend,  and  f|-(-|=L2J,  the 
(116) 


NOTATION   AND   NUMERATION.  45 

minuend ;  ^  is  the  sum  of  two  numbers  and  1| 
is  one  of  them,  and  ^g1 — J=f  |>  the  required  num- 
ber, Ans. 

Ex.37.  fxfxixfxi-W;  W+V=W; 


DECIMAL  FRACTIONS. 

NOTATION  AND  NUMERATION. 

(21O,  page  120.) 

Ex.  4.     Ans.  .496  Ex.  6.  Ans.  .0325 

Ex.  6.     Ans.  .000001  Ex.  7.  ^ks.  .0000074 

Ex.  8.     Ans.  .437549  Ex.  9.  Ans.  .3040010 

Ex.  10.  Ans.  .00000024          Ex.  11.  Ans.  .08645 

Ex.  12.  Ans.  .495705048        Ex.  13.  Ans.  .0000099009 

Ex.  14   Ans.  .04735901         Ex.  15.  Ans.  .000000000001 

Ex.  16    Ans.  .1001001001001 

Ex.  17    Ans.  .000841563436 

Ex.  18.  Ans.  .000000000000000009 


Ex.  19.  Ans    .3 

Ex.  20.  Ans.  .105 

Ex.  21.  Ans.  .0011 

Ex.  22.  Ans.  .00085 

Ex.  23.  Ans.  .100004 

Ex.  24.  Ans.  .0000704 


Ex.  25.  Ans.  46.4 

Ex.  26.  Ans.  205.65 

Ex.  27.  Ans.  60.00036 

Ex.  28.  Ans.  705.000000005 

Ex.  29.  Ans.  300.10001001 

Ex.  30.  Ans.  52.000000000005 


Ex.  31.  Twenty-four  hundredths. 
Ex.  32.  Seventy-five  thousandths. 
Ex.  33.  Five  hundred  three  thousandths. 
Ex.  34.  Seven  hundred  twenty-five  hundred-thousandths. 
Ex.  35.  Forty  million  four  hundred-millionths. 
(116-120) 


46 


DECIMALS. 


Ex.  36.  Two  hundred  fifty- six  ten-millionths. 
Ex.  37.  Ten  thousand  seventy-five  ten-millionths. 
Ex.  38.  Eight,  and  twenty-five  hundredths. 

Ex.  39.  Seventy-five,  and  three  hundred  sixty-eight  thous« 
andths. 

Ex.  40.  Forty- two,  and  six  hundred  thirty-seven  ten-thous- 
andths. 

Ex.  41.  Eight,  and  seventy-four  ten  thousandths. 
1  Ex.  42.  Thirty,  and  four  thousand  seventy-five  ten-thous- 
andths. 

Ex.  43.  Twenty-six,  and  five  hundred-thousandths. 
Ex.  44.  One  hundred,  and  one  hundred-millionth. 


Ex.1. 


.1800000 
.4560000 
.0075000 
.0000010 
.0500000 
.3789000 
.5943786 
.0010000 


Ex.1. 
Ex.3. 


REDUCTION. 


(211  ?  page  121.) 


Ex.  2.  .012000000000 
.000185000000 
.000000936000 
.000000000007 

Ex.  3.  57.300000 

900.000000 

4.755500 

100.000001 


(212,  page  122.) 
Ex.2. 
Ex.4. 
(120-122) 


=-'^  Ans. 


REDUCTION.  47 

Ex.6.     Tft0=T| 


57  l 
Ex.  10.  _JUflj{}=4, 

66| 

Ex.  11.  —  =f  $$=§, 
100 


Ex.  12.  -  =i{HJiJ==j, 
1000 

24f 

Ex.  13.  -  -=-3^5=T 
1000 

984| 
Ex.  14.  -  =i«M=H 


Ex.  15.  7T%=7|, 

Ex.  16.  24TV5^24|J,  ^n,.         Ex.  17.  f  $§**=! 

Ex.  18.        =,  ^«.  Ex.  19.  2i      = 


(314,  page  123.) 

Ex.  3.    Am     .875                 Ex.  4.    ^«s.  .56 

Ex.  5.    Ans.     .8125                Ex.  9.    Ans.  .001796875 

Ex.  11.  Ans.    .60625             Ex.  15.  Ans.  32.714286- 

Ex.  16.  Ans.    .245                 Ex.  17.  Ans.  5.783125 
Ex.  20.  Ans.    .30007 


022-124) 


48 


DECIMALS. 


ADDITION. 

(316, 

page  125.) 

Ex.1. 

.375 

Ex.  2.  5.3756 

.24 

85.473 

.536 

9.2 

.78567 

46.37859 

.4637 
.57439 

45.248377 

191.675567,  Ans. 

2.97476,  Ans. 

Ex.3. 

.5 

Ex.4.  .4675 

.37 

.325125 

.489 

.1616 

.6372 
.47856 
.02524 

.2754375 

1.2296625,  Ans 

2.50000=2.5,  Ans. 

Ex.5. 

4.65 

Ex.  6.  4.3785 

7.322 
5.3784125 

2.6487875 

2.66666+ 
5.42857+ 
12.4872 

19.9992000,  AM. 

24.9609+,  Ans. 

Ex.7. 

.137 

Ex.  8.  .0102 

.435 

.13426 

.836 

,000567 

.937 

.000003 

.496 

.24007 

2.841,  Ans. 


(125) 


.3851,    Ant. 


SUBTRACTION.  4Q 

34.72  Ex.  10.       f  f=.24743+ 

48.44  £&=.17224+ 

15217  ^=.24666+ 

95.36  TTi3F 
56.18 


.66691  db  = 

Ans.  386.87  rods.  .6669+,  Ans. 

Ex.  11.     16^=16.316—  Ex.  12.     .45 

15^=15.118—  .0275 

18if=18.484—  .009125 

14?\=14.155+  .000304 

64.07+,  Ans.  .486929,  Ans. 
Ex.  13.     1  dec.  unit  of  the  first  order=.l 

£     "      "      "     second  "     =.005 

\     "      "      "    third  «     =.00033333333+ 

\     "      «      «     fourth  «     ==.000025 

|     «      «      «    fifth  ••     =.000002 

\     "      "      «    sixth  «     =.00000016666+ 

4     «     ,«      «    seventh  «     =.00000001428+ 

Ans.  .1053605143— 


SUBTRACTION. 

(217,  page  126.) 

Ex.  4.  37.456  Ex.  5.  1.0066 

24.367  .15 


13.089,  Ans.  .8566,  Am. 

Ex  6.  1000.000  Ex.  7.  36.75 

.001  22.48 


999.999,  Ans.  14.27,  Ans. 

(125,  126) 
5 


60  DECIMALS. 

Ex.  8.     .56875  Ex.  9.     7.33333+ 

.55992  5.5625 


.00883,^/is.  1.7708+,  Am 

Ex.  10.     ff  i=.99398i9+          Ex.  11.     1. 

.000000000001 


.0491725ih,  ^ns.        Ans.    .999999999999 
Ex.  12.     57436.00         Ex.  13.     f4j|  1^4.400243+ 

If  J|f=  .227260+ 
536.74 

1756.19  4.17298+,^*. 

3678.47 

9572.15 

7536.59 

4785.94 


Ans.      29569.92  acres. 


MULTIPLICATION. 

(219,  page  127.) 

Ex.  2.     Ans.     .10464         Ex.  5.     Ans.     9.3654 
Ex.  8.     Ans.  104.976         Ex.  9.     Ans.  17.019 
Ex.  11.  Ans.     360.  Ex.  12.  Ans.     1. 

Ex.  13.  Ans.  57600.  Ex.  15.  Ans.  15.15 

Ex,  18.  Ans.  4.626  Ex.  20.  Ans.  168.48x27.375 

=4612.14  pounds,  Ans. 

Ex.  21.  2.8X36=100.8  bushels  of  oats  for  36  bushels  of 
corn;  100.8+48=148.8,  Ans. 
(126-128) 


MULTIPLICATION. 


51 


CONTRACTED   MULTIPLICATION. 


(222,  page  131.) 


Ex.2. 

36.275 

Ex.  3.      .24367 

763.4 

57.63 

1451 

731 

109 

146 

22 

17 

2 

1 

158.4±,  Ans. 

Ex.  4.      4256.785 
46500. 


548.07±,  Ans. 


8.95±,  Ans. 


Ex.  5.    357.84327 

608700.1 


21284 

357  8433 

*  2554 

25049 

170 

2863 

21 

24.008  -f-    Ans. 

AlJi 

360.6366±,  Ans. 

Ex.  6.      400.756 

Ex.7.    432.5672 

85763.1 

666660.1 

40076 

432  567 

12023 

25954 

2405 

2595 

280 

260 

20 

26 

3 

3 

461.405±:,<4n«. 


(131) 


52 

DECIMALS. 

Ex. 

8         48.4367 

Ex.  9.        7.04424=7^ 

+31531.2  =2/7 

+94658.3     =3|J| 

96873 

21133 

4844 

5635 

1453 

352 

242 

42 

5 

3 

1 

1 

103.418db,  Ans. 

27.166:4:,    ^w 

Ex 

10.      142.8373+ 

Ex.  11.      35.8756 

53025.2 

S8833.8 

28567 

28700 

7142 

1076 

286 

108 

4 

29 

1 

3 

Ans.    360.00±  degrees. 

Ans.    299.16ih  pounds. 

Ex. 

12.  ^  478.7862 

Ex.  13.      6377397.6 

"  65390.1 

643126000. 

47879 

382644 

4308 

12755 

144 

638 

24 

191 

3 

25 

A 

4 

Itui         69R  &8-I-   vnrrlfl 

Equatorial  radius,  3962.57±  miles. 


(131,  132) 


DIVISION.  63 

6356078.96 
643126000. 


381365 

12712 

636 

191 

25 

4 


Polar  radius,  3949.33±  miles. 


DIVISION. 

'(224,  page  133.) 

Ex.  4.    Am.  .2  Ex.  7.    Am.  .8666+ 

Ex.  9.    Am.  .00666+  Ex.  10.  Am.  .0075 

Ex.  11.  Am.  .0000436  Ex.  12.  Am.  .8333+ 

Ex.  13.  Ans.  .6455  Ex.  14.  Am.  6.165 

Ex.  15.  16.2-f-2.7=6,  Am. 
Ex.  16.  674-r-36.34=18.547+days,  Ans. 
Ex.  17.  5280^-14.25=370.5+,  Am. 


CONTRACTED  DIVISION. 

(226,  page  135.) 

Contracted  decimal  division  is  most  readily  performed  by 
the  method  of  inverting  the  quotient,  described  in  Note  2f 
page  135,  of  the  Arithmetic. 

(132-135^ 


54 


DECIMALS. 


Ex.  1.  4.3267)27.3782 
823.6  25960 


Ans.  6.328 ± 


1418 

1298 


120 

87 

38 
34 


Ex.3.  75.430)8.47326 
33211.  75430 


An*.  .11233 


9302 
7543 

1759 
1509 

250 
226 

24 
23 


Ex.2.  1.003675)487.24 
64.584   401 47 


Ans.  485.46± 


8577 
8029 


548 
502 

46 
40 

6 

Ex.  4.  .075637).8487564 
122.11  75637 


9238 
Ans.  11.221±:    7564 


1674 
1513 

161 

151 

10 


In  the  following  operations,  Abbreviated  Long  Division 
is  combined  with  decimal  contraction,  (see  Arithmetic,  1  IS, 
page  50). 

756.3452)8972.436 
9268.11    140898 
65263 
4756 


Ex.5.    1.436666+)478.325    Ex.6. 


249.233 


Ans.  332.942dz 


47325 

4225 

1352 

59 

2 

(135,  136) 


Ans.  11.8629± 


218 
67 


CIRCULATING  DECIMALS.  65 

Ex.  7.     1.007633)1.000000  Ex.  8.    44.736546).95372843 

524299.        93130  77813120.     589975 

2443  142610 

Ans.  ,992425±              428  Am.  .02131877+:         8400 

25  3926 

5  347 

34 

Ex.  9.     5737)4273.0  8 
8447.  2571 
276 

Ans.  ,7448±         47 
1 


CIRCULATING  DECIMALS. 
REDUCTION. 

(238,  page  139.) 

Ex.  1.  .45=£f=T5T,  Ans.        Ex.  2.     .66=f  f=f,  Am. 

Ex.  3.  .279=111=^,  Ans. 

Ex.  4.  .423=|f  f=T4T\,  Ans. 

Ex.  5.  .923076=fff^f=il,  Am. 

Ex.  6.  .9«512i=ff  ff£=f  f,  Am. 

Ex.  7.  4.72=4|f  =4T8T,  Am. 

CV     ft         9  9Q7-  _  9'297  _  911  _  85       Ana 
HiX.  O.       L.LO(  -  ^~$^~Q  -  ^37  -  3T?  ^-ns- 

Ex.  9.     2.97=2.972=2||f=2ff=i3V>,  ^TW. 
Ex.  10.    15.0=15.015=15^=15^1^,  Am. 

(239,  page  141.) 


Ex.  1.     .57=»fr«=$«=H, 
Ex.  2.     .048=^==^%=^ 
(136-141) 


56  DECIMALS. 


Ex.  3. 

Ex.  4.     .6590==«%^«t=|ff=i|,  Ans. 

Ex.  5. 

Ex.6.     .1004=^04-1  oo^V___iTVL,  AnSf 

Ex.  7. 
Ex.  8. 
Ex.  9. 
Ex.  10. 

Ex.  11.  2.029268—  ^ 


(24O,  page  142.) 

Ex.  1.     .43     ^.43333333 

.57  =-.57575757 
.4567^.45675675 
.5037^.50373737 

Ex.  2.     .578  =.57888  Ex.  3.  1.34  =1.3413413 

.37     =.37373  4.56  =4.5645645 

.2485=.24855  .341=  .3414141 
.04    =.04040 

Ex.  4.     .56T4=.5674567456745 
.34     =.3444444444444 

.247  =.2472472472472 
.67     =.6767676767676 

Ex.  5.     1.24          =1.24124124124124124 

.0578      =  .05785785785785785 
.4  =  .44444444444444444 

.4732147=  .47321473214732147 

Ex.  6.     .7          =.7777777777777 

.4567    =.4567777777777 

.24        =.2424242424242 

.346789=.3467894678946 

(141,  142) 


CIRCULATING   DECIMALS. 


57 


Ex.  7.    .8 
.36 
.4857 
.34567 


=.36363636363636363636 
=.48574857485748574857 
=.34567345673456734567 


.2784678943=.27846789432784678943 


ADDITION  AND   SUBTRACTION. 


(242,  page  143.) 


Ex,  1.  2.4444444 

.3232323 

.5675675 

7.0565656 

4.3777777 


14.7695877,  Am. 


Ex.  3.  .7854854 
.5959595 


.1895258,  Ans. 

Ex.5.  .55 
.32 
.12 

.99=1,  Ans. 


Ex.  2.  .4787878787878 
.3213213213213 
.7856485648564 
.3222222222222 
.5555555555555 
.4326432643264 


2.8961788070698,  Am. 

Ex.  4.  57.0587 
27.3131 


29.7455,  Am. 

Ex.  6.  .4387 
.8633 
.2111 
.3554 

1.8686=1.86,  Ans. 


(142, 143) 


58  <     DECIMALS. 

Ex.  7.     3.6537537 

3.1351351  Ex.  8.     .43243 

2.5646464  .25000 

.5353535 

.18243,  Ans. 
9.8888888=9.8,  An*. 

Ex.  9.  7.24574  Ex.  10.  .99000 

2.63463  .43343 


4.6lili=4.6i,  An*.  Ans.  .55656 

Ex.  11.  4.638638          Ex.  12.  :44 
8.318318  .23 

.016016 

.545454  .2i=§i=^,  Ans. 

.454545 


13.972972=13.972=13f!§=13ff,  An*. 


MULTIPLICATION   AND   DIVISION. 

(244,  page  144.) 

Ex.1.    8.4=fS;  72=31;  ^X?f=W=2.472,  Ans. 
Ex.2. 


Ex.  3.  154=411;  .2=|j  if 

Ex.  4.  4.5724=-4^W  ;  -fej  ;   - 

4^WXf-fff?=5.8793,  An*. 

Ex.  5.     4.37=W  ;  -27=^  ;  WX  1^=1.182,  An*. 
Ex.  6.     56.6=^;  ^XTi7=-i?f  =.41362530,  Ans. 
Ex.  7.    M||HXH=fi±=-7857142; 
Ex.  8.    iUlMX||=T¥A¥T 

(143,  144) 


UNITED   STATES   MONEY.  59 

Ex.9.     3.456= 


3«fc^§H3i=1.471037,  Am. 
Ex,  10.  9.17045—  V/AV  >  3.36=^V; 


Ex.  11.  §|Xf?=A65^=-  1395775941230486685032, 


UNITED   STATES   MONEY. 
NOTATION   AND    NUMERATION. 

(35O,  page  146.) 

Ex.  2.  ^rw.  $4.07        Ex.  3.  Ans.  $10.04 
Ex.  4.  .4^8.  $16.004      Ex.  5.  Ans.  $.31},  or  $.315 
Ex.  7.  ^lw*.  $1000.011     Ex.  8.  Ans.  $32.584 

Ex.  9.  Ans.  $.06i,  or  $.0625 

Ex.  10  Tweuty-one  dollars  eighteen  cents;  one  hundred 
sixty-four  dollars  five  cents;  seven  dollars  ninety 
cents ;  ten  dollars  one  cent ;  two  hundred  one  dol- 
lars twenty  cents  one  mill ;  five  dollars  thirty-seven 
and  one-half  cents;  eighty-one  and  one-fourth  cents; 
fifteen  dollars  eight  and  one-third  cents ;  ninety -six 
dollars  five  mills. 

OPERATIONS    IN   UNITED    STATES    MONEY. 

(252,  page  147.) 

Ex.  1.     $3475.50                                  Ex.  2.  $4.62} 

310.20  1.75 

1287.375  .87} 

207.125  1.00 

.62} 

$5280.20,  Ans. 

$8.87},  Ans. 
(144-147) 


60  DECIMALS. 

Ex  3.     $390.375 


$150.000                            Ex.  4.     $3800 
175.84                                              190.87} 
62.50  


8     2.035,  Ans. 


^.87},  Am 


Ex.  5.    $50.000  Ex.  6.  $  .375x150=456.25 

3.875X4    =  15.50 


$10.75 


5.50  $71.75 

2.375 


.875  $  .06}X84=    5.25 

.625  ,62}X25=  15.62} 

5.87}X2  =  11.75 


$29.875,  Ans. 

$39.12},  An* 

Ex.  7.     831.25X126.25=83945.31* 
33.75X138.25:=  4665.931 


$8611.25 

$8611.25— $6726=$1885.25,  Ans. 

Ex.  8.     $.80  X28}X40  =8  912. 

.11JX29  X300=  1000.50 
3487}x36iX20  =  2809.37} 
l2=    213.50 


811000          —  $4935.37i=$6064.62},  Ans. 

Ex.  9.     $2189.25--139=$15.75,  Ans. 
Ex.  10.  $44.748--396x$.113,  Am. 
Ex.11.  $4.50X10.75— $48.375; 

$48.375-=-7.74=$6.25,  Ans. 

Ex.  12.  84885.80-^-8287.40=17,  Ans. 
(147, 148) 


UNITED   STATES   MONEY.  Gi 

Ex.  13    $3.75+$2.875=$6.625,  cost  of  a  calf  and  a  sheep; 
$265-46.625=40,  Am. 


Ex.  14.  I  128  Ex.  15. 

9632     1730.75  41.25 
2.5 


23,  Am. 


125 
112.20 


136,  Ans. 

Ex.  16.  $2475.36  — $1936.40  =  1538.96,  the  amount  his 
money  has  diminished  since  the  beginning  of  the 
month,  which,  by  the  conditions,  must  be  f — f =f 
of  what  he  had  at  the  beginning  of  the  month. 
Hence,  $538.96-i-f=$1347.40,  Am. 

Ex.  17.  $3200— $138X12=41544,  saves  yearly; 
$1544x8=$12352,  Am. 

Ex.18.  $45.75  Xl20=$5490;  $5490— $1026=44464 ; 
$4464-f-120=$37.20,  Am. 

Ex.  19.  $6.25X425=$2656.25; 

$3088.25— $2656.25=$432 ;  $432^-$4.50=96 ; 

425+96=521  barrels,  Ans. 
Ex.  20.  $6315.12-r-36=$175.42,  wages  for  1  engineer; 

$21927.50--$175.42=125,  Ans. 
Ex.  21.  $2538+$750— $1378.56=$1909.44,  Am. 

Ex.  22.  $1.875  X-22=$41.2?;  $41.25— $25.75=$15.50  ; 

$1116-r-$15.50=72  months;  75-^-12=6  years,  Am. 

Ex.  23.  ($453.75--27.5)+$3.625=$20.125,  Ans. 

Ex.  24.  $.95-f-$1.37+$.73=$3.05,  cost  of  1  bushel  of  each 
kind;  $7G.15-:-$3.05=23  bushels  of  each  kind; 
23x3=69  bushels,  Ans. 

Ex.25.  375.5-^-2  =  187.75;  $1032.625 --187.75  =  $5.50, 
jrofit  per  acre;  $22.25+ $5.$0=$27.75,  Am 

(148,  149) 


62  DECIMALS. 

Ex.  26.  811. 374=8  V  Ex.  27.  846.75=84* 

$91  4 

2  51 


161 


187 


32  |  2093  2  I    11 


$65.40f ,  Ans.  5.5  hundred  pounds. 

Ex.  28.  $62.50—  (83.25X12|)=821.875; 
821.875-*-8.125=175  pounds,  Ans. 

Ex.29.  $10.04--16=$.6275;  $17.50--20=8>875; 

$.875  —  $.6275  =  $.2475; 
$.2475  X  320  =  $79.20,  Ans. 


PROBLEMS 
INVOLVING  THE  RELATION  OF  PRICE,  COST,  AND  QUANTITY. 


,  page  151.) 

Ex.  1.     81.32X187=8246.84,  Ans. 
Ex.  2.     $T3gX70|x5=466.09f,  Ans. 
Ex.  3.     8501.875-5-365=81.375,  Ans. 

Ex.  4.     818.48-^-8.105=176,  Ans. 

( 

Ex.  5.     $17.75-5-2=«$8.875  j 

$8.875  X  .1625  X  140=8201.906}, 

Ex.  6.     $16.50X32.40=$534.60;  Ans. 
Ex.  7.     $66.44X842.75^155992.31,  ^,s 
Ex.  8.     $10660.125-i-325.5=$32.75,  Ans. 

Ex.9.     $1.94x8.40=816.296;  812.50x1.262=815.775; 

$16.296+$15^775=$32.071,  Ans. 

Ex.  10.  837.6875  -j-81.  50=25.125=25  J  bushels,  Ans. 
(149-151) 


UNITED   STATES    MONEY. 

Dx.  11.  $3.875-h-2=$1.9375; 

$1.9375X2.172=$4.208|,  Am. 

Ex.12.  $.37£=$f;  $|X|X^ffi=*106.87i, 

Ex.  13.  $81.25--32.5=$2.50,  An*. 

Ex.  14.  $9.375X24.240=$227.25,  An*. 

Ex.  15.  $4234£-5-$5f =752$,  An*. 

Ex.16.  $20.25  X     .972=119.683       . 
2.875X15.75  =  45.281^ 
7.50  X  8.756=  65.670 


$130.634|,  An*. 

Ex.  17.  $4.625  Xl0.46=$£8.37f,  An*. 
Ex.18.  $4.70-4-2=$2.35 ;  $5.25-f-2=$2.62 £ 
$2.35  X 5.840=113.724' 
2.62^X4.376=  11.487 

$25.211,  An*. 

Ex.  19.  $2-f-$2.50=|  7ards;  Ans- 
Ex.  20.  $5.75X37=$212.75,  An*. 
Ex.  21.  $.96X38.40=$36.864,  Ans. 
Ex.  22.  $2.875 X27|X9=$715.87^,  Ans. 
Ex.  23.  $80.745-v-$.42=192|  pounds,  An*. 

Ex.  24.  $15.50  X  .327=$  5.0685 
1.625X6.72  =  10.92 
4.25  xl-108=    4.709 


$20.6975,  An*. 

Ex.  25.  $15|--$|=18,  An*. 
Ex.  26.  $5X18.962=194.81,  Ans. 
Ex.  27.  $27.90-^-15.5=$1.80,  An*. 
Ex.  28.  $125.38 X27.86=$3493.0868,  An*. 
Ex.  29.  $13.125-r-.7-^$18.75,  Ans. 
(152  i 


64  DECIMALS. 

Ex.30.  $12.75-r-2=$6.375;  $15.50-s-2=$7.75  j 
$6.375  X-  720  =$4.59 
$7.75  X.912  =  7.068 


$11.658,  Ans, 


LEDGER  ACCOUNTS. 


(258,  page  15.3  ) 

Ex.  1.  Ans.  $3434.80       Ex.  2.  Ans.  $7222.55 
Ex.  3.  Ans.  $73785.18      "Ex.  4.  Ans.  $750026.82 


ACCOUNTS   AND   BILLS. 

(267,  page  155.) 
Ex.  1.  Ex.  2. 

$2.85  X 10  =$28.50  $1.2B  Xl25=$156.25 

1.12^X16  =  18.00  1.75  X275=  481.25 

.14  X?2  =  10.08  1.121X180=  202.50 

.16^X42  ==    6.93  .87^X210=  183.75 

.40  X12  =    4.80  .84  X$0  =    67.20 

.56  X24£=  13.72  .90  X95  =    85.50 

1.06  Xl75=±  185.50 

Ans.  $82.03  30.50  X8     =  244.00 

35.75  X3    =  107.25 

.10^X958=  100.59 

.37^X40  ==    15.00 

Ans.  $1828.79 
(152-155) 


ACCOUNTS   AND   BILLS. 


65 


Ex.  3.  $27  50  X40=$1100.00 


Ex.5 


19.20  X25=    480.00 
48.10  X16=    769.60 
17.75  X12=    213.00 
26.30  X20=    526.00 
31.85  X15=    477.75 
3.87^X36=    139.50 
4.12^x42=    173.25 
2.90  X25=      72.50 

* 
Ans. 

^  $}5951.60, 

Ex.4.  $6.25  X  150=4937.50 
7.16  X275=1969.00 
5.87^170=  998.75 
1.62^X326=  529.75 
.82  X214=  175.48 
.91  X300=  273.00 
1.06  X500=  530.00 


$64.30  X24 

10.25  Xl5 

7.78  X  7 

8.45  X25 

16.12^X14 

5.90  X27 


Dr. 

=$1543.20  $17.60 
=  153.75 
=  54.46 
=  211.25 
=  225.75 
=  159.30 


$5413.48,  Ans. 
Or. 


.09^X1840=    174.80 
$2522.51 


(156,  157) 


=  156.26 

9.37^X42=  393.75 
1000.00 

3.10  X75=  232.50 
.87^X36=    31.50 

$2166.00 
2522.51 


Balance,   $356.51 


66  DECIMALS. 

Dr. 

Ex.  6.     $  .23  X§96  =$206.08 


.09^X872  = 


80.66 
54.77 
68.82 
13.50 
21.00 


,16§X81     = 
1.40  X15    = 
12£X963J=  120.42 

$565.25 


Or. 

12.25  X61=$137.25 
.22  X70=  15.40 
.87^X56=  49.00 
.68|X31=  21.31 


$222.96 
565.25 

Note  to  Bal.;     8342.29 


Ex.  1. 

Ex.  2. 
Ex.  3. 
Ex.  4. 

Ex.  5. 


Ex.  6. 

Ex.  7. 


Ex.  8. 
Ex.9. 


PROMISCUOUS   EXAMPLES. 

84.875  Xl2f=$61.54+,  Ans. 
$33.75-4-$.375=90;  90-^-2^=36,  Ans. 
36X36=1296;  $97.20-f-1296=$.075,  Ans. 

$5.3.5-r-.625=$8.56,  Ans. 

.0|  =.033| 
.00|=.008| 

Or,  iS—  rfu=A=- 


.025  ,  Ans. 

8142^x26^f=814.495x26.46875 

=21558.66+,  Ans> 
$75X5=«375;  $68x12=8816;  5+12=17; 

$375  +  8816  +  $118  =  $1309  ; 
$1309--17  =  $77,  Ans. 

$.625  X  -8=4-50,  Ans. 
$.87^  +  $.18|  +  $.10|  =$1.165; 
$27.96  -r-  $1.165=24,  Ans. 
(157,  158) 


PROMISCUOUS   EXAMPLES.  67 

Ex.  10.  13543.47--365.25=37.08  miles  in  1  day; 
37.08X1=32.445  miles,  Am. 

Ex.11.  $5.12;|X100=$512.50       $6.50  X  75=4487.50 
1.06^X250=  265.62^       1.37£X250=  343.75 

221.874  — •• 

$831.25 

$1000.00  — 8831.25=8168.75,  to  be 
realized  on  the  remaining  25  barrels  of  flour;  hence 

3.75--25=$6.75,  Ans. 


Ex.  12.  4580.289-4-114.45=40.02  bushels  from  1  acre; 
120.06-5-40.02=3  acres,  Ans. 

Ex.  13.  .017226+-r-.030625=.5625ib,  Ans. 
Ex.  14.  13.5--.0225=600,  Ans. 

Ex.  15.  8.5-4-5=1.7  rods,  his  daily  work; 

59.5—8.5=51 ;  51-^-1.7=30  days,  Ans. 

Ex.  16.  «.375X28.5=810.68|;  12520--2000=6.26  tons; 
.75   X4.53=    3.391 


$14.085-r-6.26=$2.25,  Ans. 

Ex.17.  1826+1478+1921=5225;  $8.80-5-2=; 

$4.40X5.225=$22.99    $.09x31=82.79 

5.25X2.81  =  14.751  4.50x6^=29.25 

$37.74^        —       $32.04=85.70 J,  Ans. 

Ex.18.  $122.50-v-35=$3.50;  $3.50X29=$101.50,  Ans. 

Ex.19.  $.56|X1200=$675 

168.675 


$843.675 
$.60X375.5=  225.30  1200—375.5=824.5; 

8618.375-5-824.5=8.75,  Ans. 
(158, 159) 


Ex.  20. 


1680 


DECIMALS. 

$2.856  Ex.  21.        8 

2000  .125 


$127 
25.42 


$3.40,  Ans.  $3228.34,  Ans. 

Ex.  22.  ($100X150)—  $3900=411100,  cost  of  whole; 
$11100-;-150—  $74,  cost  per  acre; 
$11100—  $2250—  $8850,  sold  for; 
$8850—150—  $59,  sold  for,  per  acre. 

Ex.  23.  $14.375X212.5—  $3054.68|,  cost; 
1.75  X  2125—  $3718.75,    avails; 

$664.06|,  Ans. 

Ex.  24.  $545-r-10—  $54.50,  cost  of  1  acre; 
$17712.50--$54.50—  325,  Ans. 

Ex.25.  224.56x7XiXi=196.49,^7is. 
Ex.  26.  $169.8125—  $39.1875—  $130.625; 
$130.625-^-104.5—  $1.25,  Ans. 

Ex.  27.  $6.975-r-.93—  $7.50,  Ans. 
Ex.28.  $4000X.375X-12=$180,  Ans. 

flf       2^ 
Ex.29.      —  -—     XfXi=3XlX|X$X|Xi 

[4|      2i  J  =^=.15,  Ans. 

Ex.30.  $.331X375—  $125;  8125-5-7.5=  $16.66|,  Ans. 

Ex.  31.  1-|-.  84—  1.84  times  the  sum  invested; 

1.84x2=3.68  times  the  sum  invested,  Ans. 

Ex.32.  T%=.3;  f—  .6;  1—  (.3+.6)=  .1.  Now,  if  he  pur- 
chases .3  of  a  bushel  of  barley,  .6  of  a  bushel  of 
wheat,  and  .1  of  a  bushel  of  oats,  he  will  have  1 
bushel  of  grain,  worth 

$  .625X-3=$  .1875 
1.875X-6=  1.125 
1=    .0375 


$1.35 
(159,  160) 


CONTINUED   FRACTIONS.  09 

And  for  $54,  he  can  purchase  as  many  bushels  as  $1.35  is 
contained  times  in  $54.     Therefore, 
|54-f-81.35=40,  Am. 

Ex.33.  191X27=522  yards; 

$4.311  X522=$2251.12£ 
881.871 

9.621 

$2642.62i-f-522-j-$5.06|,  An*. 

Ex.  34.  $1.18f     1356  $.41     736 

•1.12        870  .31     528 


$  .061  x  486=$30.37i  >         $.10x208=$20.80  ; 
$30.37£+820.80+«13.62£=$64.80,  entire  loss; 
$235.87|—  $64.80=$171.07|,  gained,  Arts. 

Ex..35.  j?  i+4J|=  |;  |  -*-2=A,  greater; 


Ex.  36.  1  +  !=   y  times  his  original  capital,  end  of  1st  year  ; 
is  x  |=  If     "         "         "         "         "     2d     « 

||Xl|=Vo3     "         "         "         "         "     3d     a 
$28585.70--  ^=$17991,  his  original  capital; 
$28585.70—  $17991=$10594.70;  gain,  Ans. 


CONTINUED  FBACTIONS. 

(»71,  page  162.) 

The  division  may  be  performed  in  the  same  manner  as  in . 
finding  the  greatest  common  divisor.     (See  loO,  Higher 
Arithmetic.) 

(160-162) 


70 


CONTINUED   FRACTIONS. 


1240 
1042 

Es 

5 
2 
2 
1 
1 
1 
2 
2 
10 

Ex 
516! 

447 

..  2. 
6721    1 

Ans.  
6200    5+1 

521       2+1 

198 
125 

396        2+1 

125           1+1 

73 
52 

73             1+1 

52               1+1 

21 
20 

42                 2+1 

10                   2+1 
10                     10 

.3. 

501    1 

-A.71S 

223874 
207459 

1 

2 

3 
4 
4 
1 
2 
3 
16 
3 

748    2+1 

69153      3+1 

16415 
13972 

65660         4+1 

3493           4+1 

2443 
2100 

2443            .  1+1 

1050               2+1 

343 
336 

1029                 3+1 

21                   16+1 
21                      8 

7 

(162) 


REDUCTION. 


71 


Ex.4. 


121 

Av 

1 

29     4 

116 

'  4+1 

25     5 

5 

5+1 

4     1 

4 

1+1 

4    4 

1 

4 

(273,  page  163.) 

Ex.  1.     Terms  of  continued  fraction,  ^,  -J, 
approximate  values,  £,  f,  ^§,  T%\, 

Ex.  2.     Terms  of  continued  fraction,  |,  |,  1, 

approximate  values,  J,  T%,  ^5T,  ^,  B8 
Ex.  3.     Terms  of  continued  fraction,  ^,  A,  | ; 

approximate  values,  4;  3%?  fVy;  ^ws- 
Ex.  4.     Terms  of  continued  fraction,  *,  ±,  ^,  | 

approximate  values,  ±,  y\?  A?  7%%  T45 
Ex.  5.  Terms  of  continued  fraction,  y>  3,  •}>  -g  j 

approximate  values,  1,  f ,  T9g,  ||,  Ans. 


COMPOUND  NUMBERS 

REDUCTION  DESCENDING. 

(367,  page  194.) 

Ex.  2,    £133x20+6  s.=2666  s. ; 

2666  S.X12+8  d.=32000  d. ; 
32001  d.x4=128000  far.,  Ans. 
(162-194) 


72  COMPOUND   NUMBERS. 

Ex.  3.     100  mi. X 63360=6336000  in.,  An*. 

NOTE. — When  the  number  given  for  reduction  contains  but 
one  denomination,  the  scale  of  relation  may  be  taken  from  the 
table  of  Unit  Equivalents,  and  the  answer  obtained  by  a  single 
operation. 

Ex.  4.     1|  mi.x4=6  mi.,  length  of  fence; 
6  mi.  X 320=1920  rd.,  Ans. 

Ex.  5.     8X3X3=72  cu.  ft.,  solid  contents  of  the  block; 

175  Ib.X  72=12600  lb.;  12600  lb.-^-100=126cwt.; 
126  cwt.-r-20=6  T.  6  cwt.,  Ans. 

Ex.  6.     1  hhd.=63  gal.;  $.28x63=$17.64,  Ans. 

Ex.  7.     1548  bu.  1  pk.=6193  pk. ;  2  bu.  3.  pk.=ll  pk. ; 
6193-*-ll=563,  Ans. 

Ex.  8.     $3.75xlO=$37.50;  10  bu.=640  pt.; 

$.06|X 640=$40.00;  $40— $37.50=$2.50,  Ans. 

Ex.9.  90°X60+17'=5417';  5417' X 60+40" 

=325060,"  Ans. 

Ex.  10.  The  18th  century  embraced  the  time  from  the  com- 
mencement of  A.  D.  1701  to  A.  I}.  1800  inclusive, 
and  1800  was  not  leap  year;  hence,  100  yr.X365 
+24  da.=36524  da.,  Ans. 

Ex.  11.  lgreat-gross=1728  units ;$.06|X  1728=$108,  Ans. 

Ex.  12.  4  bales  4  bundles  1  ream  10  quires=990  quires; 

24-r-8=3  vol.  per  quire;  990x3=2970  vol.,  Ans. 

Ex.  13.  18  yr.x365+24  da.=6594  da.; 
6594  da.  X  24=158256  h.; 
158256  h.x60=9495360  min.,  Ans. 

Ex.  14.  481  sov.X240=115440  d.,  Ans. 
Ex.  15.  $7f =$7.375;  $7.375x1000=7375  mills,  Ans. 
Ex.  16.  3  P. X  130=390  gal.;  390  gal. X 4=1560  qt.,  An*. 

(194) 


REDUCTION.  73 


Ex.  17.  37  ellsX5+l  qr.=186  qr.; 

186  qr.-^-4=46  yd.  2  qr.,  Ans. 

Ex,  18.  £6  10s.  10d.=1570d.; 

$.02^X1570=$31.66+,  Ans. 

Ex.19.  60.Xl6+14fg=110fg; 


883  £3X60+45^=53025^,  Ans. 

Ex.  20.  1  T.  1  P.  1  hhd.=7  hhd. ; 

7  hhd.X  52^=367^  gal,  Ans. 
Ex.  21.  £126  12  s.  6  d.=30390  d.; 

30390  d.-7-60=$506£,  Ans 
Ex.  22.  1  hhd.=504  pt.;  2  qt.+l  qt.+l  pt.==7  pt.; 

Ex.  23.  2  ft.  9  in.  =  33  in.;  63360^-33  =  1920  steps  in  1 
mile;  1920x95=182400,  Ans. 

Ex.  24.  $lf  Xl2=$21,  cost;  12  bbl.Xl26=1512  qt.; 

$.06X1512=190.72;  $90.72— $21=$69.72,  Ans 

Ex.  25.  75  A.  X 10+4  sq.  ch.=754  sq.  ch.; 
754  sq.  ch.X  16+18  P.=12082  P.; 
12082  P.X625+118  sq.  1=7551368  sq.  1.,  Ans. 

Ex.  26.  4  in.  X  16=64  in.,  Ans. 

Ex.  27.  150  leaguesX3Xl-15=517.5  miles,  Ans. 

Ex.  28.  (50  A.— 14  A.)  X  160=5760  sq.  rd.,  Ans. 

Ex.  29.  36  Ib.  8  oz.=8800  pwt.; 

$1.042X8800=89169.60,  Ans. 

Ex.30    9  cwt.  42  lb.=9421b.; 

942  Ib.X 8=7536  Ib.;  7536^-48=157,  Ans. 

Ex  31.  $1|X12= $15,  cost;  12  bbl.X280=3360  Ib.; 
$.0075X3360=$25.20; 

$25.20— $15=110.20,  Ans. 

(194,  195) 

7 


74  COMPOUND   NUMBERS. 

Ex.  32.  1  Ib.  10  g^22g=10560  gr.=1056  doses; 
$2.25X22=$49.50;  $.1 

$132— $49.50=882.50,  An*. 


(368,  page  196.) 

Ex.1. 
Ex.2. 
Ex.3. 
Ex.4.  -312  gal.XfXfXf  = 

E*.  5. 

Ex.6. 
Ex.7. 

Ex.8. 

Ex.  9.     f  X  T4T  rd.  X  V  =  V2  Jd-> 

Ex.  10.  t%  wk.Xf  Xf=8j  da.,  Ans. 

EX.  11. 


(369,  page  197.) 

Ex.1.     T9o  yd.X3=2T7o  ft.;   ^  ft.xl2  =  8f  in.; 

Ans.  2  ft.  8|  in. 

Ex.  2.     |  mo.X30=24  da.?  Ans. 

Ex.  3.     ||§  T.X20=18j^  cwt.;  |4  cwt.XlOO=96f  Ib.; 

|  lb.Xl6=14  oz.,;  Ans.    18  cwt.  96  Ib.  14  oz. 

Ex.  4.     f  T.X20=11^  cwt.;  £  cwt.x4=|  qr.; 
|  qr.X28  =  12|  Ib.;  |  lb.Xl6=7^  oz.; 

JL?is.  11  cwt.  12  Ib.  7-g  oz. 

Ex.5.     ?X41 


(195—197) 


REDUCTION.  75 

Ex.  6     &  A.X4=2T^  R;  ^ 


iJsq.  yd.x9=  5ff  sq.  ft.; 
||  sq.  ft.Xl44=127T5y  sq.  in.; 

.4w«.  2  E.  6  P.  4  sq.  yd.  5  sq.  ft.  127^  sq.  in. 
Ex.  7.     I  mi.x8=3s  fur.;  f  fur.x40=174  rd.; 

2i         21 
4rd.Xl6£=2—  ft.;  —  ft.Xl2=4f  in; 

.4n*.  3  fur.  17  rd.  2  ft.  4|  in. 

Ex.  8.     4  great-gross  Xl2=6f  gross;  f  gross  X  12=1  Of  doz.; 
§  doz.  X  12=3  1  units;  ^tws.  6  gross  10  doz.  3|  units. 

Ex.  9.     T9g  great  ciro^X  360x60=12150  mi.;  Am. 

Ex.  10.  V  Cd.X|=2A  Cd.;  ^  Cd.x8=5T%  cd.  ft.; 
T^  cd.  ft.  X  16=9  1  cu.  ft.; 

Ans.  2  Cd.  5  cd.  ft.  9f  cu.  ft. 

Ex.  11.  438  mi.Xi=262|  mi.;  |  mi.x8=6f  fur.; 

f  fur.x40=16  rd.;          Ans.  262  mi.  6  fur.  16  rd. 
Ex.12.  f|f  3X8=3/2  f  3;  &f  3X60=35  TIL; 

Ans.  3  f  ^  35  T^ 

Ex.  13.  |  S.X30—  12f°;  $0x60=51f';  f  X60=25f"; 

^iw.  12°  51'  25f". 

Ex.  14.  T^  hhd.X63=33l|  gal;  if  gal.x4=3T^  qt.; 
JL  qt.X2=!T53  pt.;  A  P^X4=1T73  gi.; 

^Ins.  33  gal.  3  qt.  1  pt.  1T^  gi. 

(370,  page  198.) 
Ex.  1.     .645  da.X24=15.48  h.;  .48  h.x60=28.8  min.  ; 

.8  min.x60=48  sec.;      Ans.  15  h.  28  min.  48  sec. 

Ex,  2    .765  Ib.x  12=9.18  oz.  ;  .18  oz.X  20=3.6  pwt.; 

.6  pwt.x24=14.4  gr.        Ans.  9  oz.  3  pwt.  14.4  gr. 
(197,  198) 


76  COMPOUND   NUMBERS. 

Ex.  3.     .6625  mi.  X?  =5.3  fur.;  .3  fur.x40=12  rd.; 

Ans.  5  fur.  12  rd. 
Ex.4.     ,8469°X60=50.814';  .814'X60=48.84;" 

Ans.  50'  48.84". 

Ex.  5.     .875  hhd.X  63=55.125  gal.;  .125  gal.x8=l  pt.; 

Ans.  55  gal.  1  pt. 

Ex.  6.     £.85251X20=17.0502  s.;  .0502  s.Xl2=.6024d.; 
.6024  d.x4=2.4096  far.;  Ans.  17  s.  2.4+far. 

Ex.  7.     .715°  X  60=42.9';  .9'X60=54";        Ans.  42'  54". 

Ex.  8.     .88125  A.X4=3.525  K;  .525  R.X40=21  P.; 

Ans.  7  A.  3  R.  21  P. 

Ex.  9.     .625  fath.x6=3.75  ft.=3|%,  Ans. 

Ex.  10.  .375625  bbl.X200=75.1251b.;  .125  Ib.X  16=2  oz.; 

Ans.  75  Ib.  2  oz. 

Ex.  11.  .1150390625  Cong.x8=.9203125  0.; 


8  f  5X60=48  nt;  Ans.  14  f  §  5f  5  48  i^. 

Ex.  12.  .61  tunX2=1.22  P.;  .22  P.X2=.44  hhd.; 

.44  hhd.X  63=27.72  gal.;  .72  gal.x4=2.88  qt.; 
.88  qt.X2=1.76  pt.;  .76  pt.x4=3.04  gi.; 

Ans.  1  P.  27  gal.  2  qt.  1  pt.  3.04  gi. 


REDUCTION   ASCENDING. 


(371,  page  200.) 

Ex.  1.     1913551  dr.-f-16=119596  oz.  15  dr.;  119596  oz.-- 
16=7474  Ib.  12  oz. ;  7474  lb.-f-2000=3  T.  1474  Ib.  ; 
Ans.  3  T.  14  cwt.  74  Ib.  12  oz.  15  dr. 
(198-200) 


REDUCTION.  77 

Ex.  2.  97920  gr.--20=4896  sc. ;  4896  sc.-^3^1632  dr.; 
1632  dr.-i-8=204  oz. ;  204  oz.-T-12=17  lb.,  Am. 

Ex.  3.  1000000  in  --12=83333  ft.  4  in.; 
83333  fW3=27777  yd.  2  ft.; 
27777  yd.-r-5,|=5050  rd.  2  yd.; 
5050  rd.--40=126  fur.  10  rd.; 
126  fur.--8=15  mi.  6  fur.; 

Ans.  15  mi.  6  fur.  10  rd.  2  yd.  2  ft.  4  in, 

Ex  4.     120X56=6720  sq.  rd.; 

6720  sq.  rd-^160=42  A.,  Ans. 

Ex.  5.     60  X 15  X  10=9000  cu.  ft. ; 

9000  cu.  ft.-4-16=562  cd.  ft.  8  cu.  ft.; 

562  cd.  ft.-*-8=70  Cd.  2  cd.  ft. ; 

Ans.  70  Cd.  2  cd.  ft.  8  cu.  ft. 
Ex.  6.     28  ft.  6  in.=342  in. ;  6  ft.=72  in. ; 

342--72=4f  fath.,  Ans.     Or, 

28  ft.  6  in.=28!  fM  28£-s-6=4f  fath.,  Ans. 

Ex.  7.    30876  gi.-s-4=7719  pt. ; 

7719  pt.^-2=3859  qt.  1  pt.; 
3859  qt.^-4^964  gal.  3  qt. ; 
964  gal.-=-63=15  hhd.  19  gal.; 

Ans.  15  hhd.  19  gal.  3  qt.  1  pt. 

Ex.  8.     27072  qt. -^8=3384  pk.;  3384  pk.-^-4=846  bu.,  Ans. 

Ex.  9.  254-^-2=127  gi.;  127  gi.-?-4=31  pt.  3  gi.; 

31  pt.-i-2==15  qt.  1  pt.;  15  qt.-^4=3  gal.  3  qt.; 
Ans.  3  gal.  3  qt.  1  pt.  3  gi 

Ex.  10.  1234567  far.-f-4=308641  d.  3  far.; 
308641  d.-5-12 =25720  s.  Id.; 
25720  s.-^-20=£1286;  Ans.  £1286  1  d.  3  far. 

Ex.  11.  One  half  crown=2  s.  6  d.=30  d.; 
2468H-30=82-r£  half  crowns,  Ans 
(200) 


78  COMPOUND   NUMBERS. 

Ex.  12.  $88.350-v-$.186=475  francs,  Am. 

Ex.  13.  622080  cu.  in.--1728=360  cu.  ft.; 
360  cu.  ft.--40=9  T.,  Am. 

Ex.  14.  84621  TTL  --60  =  1410  f  3  21  r^  ;  1410  £5 -5-8  = 
176  f  I  2  f  3 ;  176  f  2-5-16=11  0. ;  11  0.--8=1 
Cong.  3  0.;  Am.  1  Cong.  3  0.  2  f  3  21  TV 

Ex.  15.  135000000-r-1728=78125  great-gross,  Am. 

Ex.16.  1020300"  -5-60  =  17005';  17005'  -f-  60=283°  25'; 
283°-^30=9  S.  13°;  Am.  9  S.  13°  25'. 

Ex.  17.  411405  sec.-,- 60=6856  min.  45  sec. ;  6856  min.-s- 
60=114  h.  16  min. ;  114  h.--24=4  da.  18  h; 

Am.  4  da.  18  h.  16  min.  45  sec. 

Ex.  18.  412'--60=60  52',  Am. 

Ex.19.  360°X60  =  21600';  21600' X 20  =  432000  min.  of 

time;  432000  min. -^60  =  7200  h. ;  7200  h.-s-24 

300  da.,  Am. 

Ex.20.  120X144=17280;  17280-^20=864,  Am. 
Ex.  21.  180°X69.16=12448.8  mi.,  Am. 

Ex.  22.  45  min.-)-25  min. =70  min.  gained  each  day; 

36  yr.X365+9  da.=13149  da.; 

13149  da.  X  70=920430  min. ; 

920430  min.-f-60=15340  h.  30  min. ; 

15340  h.--24=639  da.  4  h. ; 

Am.  639  da.  4  h.  30  min 
Ex.  23.  20X4=80  qt.  bought.; 

20 X 282=5640 cu. in.;  5640-^-57|=97f4  qt.  sold; 

97^4  qt.— 80  qt.=17f  4  qt.  gained,  Am. 

Ex.  24.  1500  bu.X 35=52500  lb.; 

52500  lb.-~28=1875  bu.,  Am. 

Ex.  25.  120  lea  X^X  1.15=414  mi.,  Ans. 
(200,  201) 


REDUCTION.  79 

Ex.  26.  1  bbl.  1  gal.  2  qt.=33  gal. ; 
33  gal. X 231=7623  ou.  in.; 
7623  cu.  in.-r-282=271533:  beer  gal.,  Ans. 

Ex.  27.  150  bu.X 2150.4=322560  cu.  in.; 

322560-^-2218.2=145.415+  Imp.  bushels,  Ans. 

Ex.  28.  68  ft.  8  in.=68£  ft.;  68f  X33=2266  sq.  ft.; 
2266  sq.  ffc.-5-100=22f-g  squares,  Ans. 

Ex.  29.  4  ft.=48  in.;  3  ft =36  in.;  1  ft.  6  in.=18  in.; 
48X36X18=31104  cu.  in.,  Ans. 

Ex.  30.  120X56=6720  P.;  6720  P.-*-160=42  A.,  Ans. 

Ex.  31.  856  dr.x3X20=21360  gr.; 

21360  gr.-f-24=890  pwt.;  890  pwt.-=-20=44  oz.  10 
pwt.;  44  oz.-^-12z=3  lb.  8  oz.; 

Ans.  3  lb.  8  oz.  10  pwt. 

Ex.  32.  175T.x2240=3920001b.;  392000--2000=196T., 

short  ton  weight;  $3.75  X  175  =  $656.25,  cost; 
$4.50X196=J882,  sold  for; 
$882— $656.25=4225.75  gain,  Ans. 

Ex.  33.  73750-^-1.25=59000  sq.  ft.; 

59000  sq.  ft.-5-272|=216  P.  194  sq.  ft;  216  P. 
-^160=1  A.  56  P.;        Ans.  I  A.  56  P.  194  sq.  ft 

Ex.  34.  2492  lb.-^-56=44.5  bushels  of  corn; 

2175  lb.-f-60= 36.25  bushels  of  wheat; 
$.60X44.5=826.70;  $1.20x36.25=843.50; 

$26.70+$43.50=$70.20,  Ans. 


Ex.  39. 


72 


75  Ex.  40. 

42 


4  I  175 


90 


5 

6 
76 


3  |  76 
5;  Ans. 

Ans.  $25.33^ 
(201-203) 


80 
Ex.  41. 


COMPOUND  NUMBERS. 


21 


2 
126 


Ex.  42.  Myxf-*)=$52.50j  rated  at  in 
Vermont ; 

*  $52.50+$7.50=$60,  to  be 

24  lb.,  Ans.  sold  for  in  New  Jersey ; 


(372,  page  203.) 
Ex.1.     *  s.X^=£^>  A™. 
Ex.  2.     f  pwt.X31TXT13=?iff  lb->  ^TW 
Ex.  3.     |  lb. 
Ex.4.     fX 
Ex.5.        pt. 


Ex.  6.    ^I^XlXf  pt.XiXi=|  pk-=§  of  2  pk., 
Ex.7.     |X|XT\cd.  ft.x|=- 

Ex.  8.     T%XT4?X  V  P.XT|o 

Ex.  9.     |X  V  fur.  X  1=11  mi.;  and  ||  mi.  is  ^  of  T\j  of 

||XfX  T2  nd-=12f  mi.,  -4rw.     Or, 

IX  y  XlXfXV2  mi.=12|  mi., 
Ex.  10.  |Xf  X^F  cu.  ft.X^=it 
Ex.  11.   \f  Xf  X  V  cu-  ft-XTi8- 
Ex.  12.  f  in.X^V^g1^  E-  E-?  Ans- 


(3739  page  205.) 

Ex.  1.     2  K.  20.  P.^100  P.  ;  1  A.=160  P.; 

i  oo   A  _  5   A      An* 
T50  A  —  8  **^  -o-WS. 

Ex.  2.     6  fur.  26  rd.  3  yd.  2  ft.=4400  ft.  ;  1  mi.=5280  ft.; 


Ex.  3     18  s.  5  d.  2T23  far.=  U_5JLP  far;  £lz=960  d.  ; 

T=4|,^ 
(203-205) 


REDUCTION.  81 

Ex.  4.     7  I  7  3  2  9  14  gr.=3834  gr. ;  21  ft.  =  120960  gr.; 


Ex.  5.     4  da.  16  h.  30  min. = 6750mm;  3  wk.= 30240  min.; 


Ex.  6.     —  bu.—  T7g  bu.,  Ans. 

Ex.  7.     28  gal.  2  qt.  =  114  qt.  ;  1  hhd.=252  qt.  ; 

252  qt.—  114  qt.  =  138  qt.;  ^||=f  |,  Ans. 

Ex.  8.     4  bundles  6  quires  16  sheets:=4000  sheets  ; 

1  bale=4800  sheets;  |§f8bales=|  of  a  bale,  Ans. 

Ex.9. 


Ex.  10.  $—  = 


=$6^=$6.30,  Ans. 


Ex.  11.  3  0.  3  f  g  1  f  3  36  nt  =24576  nt  ; 

1  Cong.=61440  nt  ;  ||||^  Cong.=|  Cong.,  Ans. 


Ex.  12.  36  cu.  ft.  864  cu.  in.=63072  cu.  in.; 
1  T.=50  cu.  ft.  =86400  cu.  in.; 


Ex  1.    60 

60 
24 

7 

(374, 
48.0    sec. 

page  206.) 
Ex.  2.     60 
60 
30 

46.44" 

46.80  min. 

27.774' 

9.78  h. 

3.4629° 

5.4075  da. 

An*.           .11543  S. 

^Ins.           .7725  wk. 

(205,  206) 


82 


COMPOUND  NUMBERS. 


Ex.3.    40 
4 

11.52  P. 

Ex. 

.     3.25  I 

7.     16 
10 
23040 

4.     24     19.2  gr. 

20     16.8  pwt 

3r.288  E. 

4       2.84  oz. 

Ans. 
Ex.  5.     12 
3 
5.5 
40 

.322  A. 
11.04  in.    Ex.6 

Ans.      .71 
5-f-12=.27083  ft. 
12.00  P. 

1.92  ft.        Ex. 

2.64  yd. 

4.75  sq.  ch. 

28.48  rd. 

126.475  A. 

Ans.        .712  fur. 

Ans. 

.0054893+Tj, 

Ex.  8.     3.75  ft.H-6=.625  fath., 

Ex.  9.     .45  pk.-=-4=r.ll25  bu. ;  .1125--1.25=.09,  Ans. 

Ex.  10.  3  A.  2  E.=560  P.;  IE.  11.52  P.=51.52  P.; 

51.52-*-560=.092,  Ans. 
Ex.11. 


Ex.  12.  63     36.00  HI 

Ex.  13.       2 
4 

252 

1.0  pt. 

8       5.6  f  % 

3.5  qt. 

Ans            .7  f  § 

50.875  gal. 

Ans.             .20188+T. 

ADDITION. 

(377,  page  208.) 

Ex.  3.     Ans.  3  ini.  2  fur.  27  rd.  16  ft. 
Ex.  4.     Ans.  1017  A.  2  E.  36  P.  15  sq.  yd.  5  sq.  ft.  72  sq.  in. 
Ex.  7.     Ans.  15  Cd.  4  cd.  ft.  4  ou.  ft. 
(206—208) 


ADDITION.  83 

fix,  8.  1|  hhd.         =1  hhd.  42  gal. 

42  gal.  3  qt.  1|  pt.=      42  «  3  qt.  1  pt.  1  gi. 
|  gal.         =          3  "  1  « 
2  qt.  |  pt.       =          2  "  0  "  3  " 
1.75  pt.        =  1  «  3  «  . 


*.  2  hhd.  23  gal.  2  qt.  0  pt.  3  gi. 

Ex.  9.  145|  A  =145  A.  3  E.  20  P. 

7  "  2  "  29|  " 

1  «  3  "  16£  " 

A.=      3  "  13  « 


156  A.  0  R.  39|  P.,  Ans. 

Ex;  10.  31  bu.  2pk. 

10|  bu.      =10  «  3  "  4  qt. 

5  bu.  6£  qt.   =  5  «  0  «  6  «  1  pt. 

14  bu.  2.75  pk.=14  "  2  «  6  « 


62  bu.  1  pk.  5  qt.  If  pt.,  Am. 

Ex.  11.  42  yr.  7£  mo.    =42  yr.  7  mo.  15  da. 
10  yr.  3  wk.  5  da.  =10  "  0  "  26  « 
9|  mo.  9  «  22  «  12  h. 

1  wk.  16  h.  40  min.=         7  "  16  "  40  min. 
§  mo.         =        25  « 
3|  da.        =         3  «  19  «  12  min. 


s.  53  yr.  7  mo.  9  da.  23  h.  52  min. 

Ex.  13.  1£  gross  1\  doz.=  304 

8   «  1|  «  =  453 

|  great-gross  =1296 

6|  doz.      =  75 

4  doz.  7  units  =  55 


(208,  209) 


84  COMPOUND  NUMBERS. 

Ex.  15.     3|  Pcli.  18  cu.  ft.=4  Pch.  1  en.  ft.  864    cu.in. 

84.6  cu.  ft.  =3     «  10  "       604.8    « 

§  Pch.  =          20  "     1080        " 

f  o  cu.  ft.  =  1280        « 


Ans.  8  Pch.  9  cu.  ft.     804.8    cu  ID 


Ex.  16,  $  3.75 
25.50 
12.875 

2.40 
2.5475 


$47.0725,  Ans. 

Ex.  18.                     42.4  bu. 

2866  lb.=  49.414-  « 

36|  bu.  =  36.75  " 

39  bu.  29  lb.=  39.5  " 


$.60X168.063=4100.84-,  Ans. 

Ex.  19.  1.125  T. 

If  T.=1.4      « 
2500  lbs.=1.25    « 

$8X3.775=$30.20,  Ans. 

Ex.20      140|cu.  yd.  =140.8      cu.  yd. 

24.875      " 
46  ou.  yd.  20|  cu.  ft.=  46.75        « 

212.425  cu.  yd.  remove*  , 
$.18X212.425=$38.24— ,  cost. 


(209) 


SUBTKACTION.  85 


SUBTRACTION.  •      - 

(379,  page  211.) 
Ex.  3.  Am.  2  hhd.  54  gal.  If  qt. 
Ex.4. 

45  yr.  1  mo.  3  wk.  0  da.  17£  h.=45  yr.  1  mo.  21  da.  17.5  h. 
10  «  9  «  1  "  22  "  6.8  "=10  "  9  "  29  «'  6.8  h. 

Ans.  34  yr.  3  mo.  22  da,  10.7  h. 
Ex.  6.  Ans.  12  cwt.  85  Ib.  6  oz. 

Ex.  7.  2  wk.  3|  da.=2  wk.  3  da.  20  h. 

.659  wk.   =    4  "  14  «  42  min.  431  Seo. 


Ans.    1  wk.  6  da.    5  h.  17  min.  16|  sec. 

Ex.  8.     ££|   hhd.=32.90625  gal. 
.90625    « 


32  gal., 

Ex.  9.  |  of  3|  A.=l  A.  1  K.  20   P. 
3  "  12.56  " 


2  E.   7.44  P.,  Ans. 

Ex.  10.        10  Ib.  8  oz.  8  pwt. 
D>-=         18  " 


10  Ib.  7  oz.  10  pwt., 

Ex.  11.  36  Cd.  4  cd.  ft. 

10  "  6   "  12  cu.  ft. 


25  Cd.  5  cd.  ft.  4  cu.  ft.,  Ans. 

Ex.  12.  5£  bbl.=5  bbl.  15  gal.  3  qt. 
4hlid.=l  "   4  «  2  « 

4  bbl.  11  gal.  1  qt., 

(211) 


86  COMPOUND  NUMBERS. 

tfk.=4  da.  16  h. 

da.=          15  "  10  min.  30  sec. 


Ex.  13.  |  wk.=4  da.  16  h. 


4  da.    0  h.  49  min.  30  sec.,  Ans. 

i  ' 

Ex.  14.  |  gross =7^  doz.; 

7^  doz. — |  doz. =6 1  doz.,  Ans. 

Ex.  15.  I  mi.=6  fur.  8  rd.  4    yd.  2  ft.    8  in. 
rd.=  5     "    0  «     9  " 


6  fur.  7  rd.  4£  yd.  1  ft.  11  in.  ; 
Or,          6  fur.  7  rd.  5    yd.  0  ft.    5  in.,  Ans. 

Ex.  17.  f  pk.=6  qt.;  .0625  bu.=2  qt.  ; 
6  qt.  —  2  qt.=4  qt.,  Ans. 

Ex.  18.  §  of  365|  da.  =  28  wk.  6  da.  22  h. 
of    5    wk.=  4   "    1  «      4  « 


33  wk.  1  da.    2  h.—  49|  min.= 
33  wk.  1  da.    1  h.  lOf  min., 

Kx.  19.  f  of  3|  mi.+174  rd.  =  l  mi.  1  fur.; 
1  mi.  1  fur.—  5|  fur.—  3|  fur.,  Am. 

Ex.  20.  15  bbl.  3.25  gal.  =15  bbl.    3    gal.  1       qt. 
14    «    24      «    3.54  " 


9^  gal.  1.46  qt. ; 
Or,    9    gal.  3.46  qt.,  Ant. 

Ex.  21.  1457  lb.+1578  lb.+1420  lb.=4455  Ib. 
=92  bu.  39  Ib.;  200  bu.— 92  bu.  39  Ib. 
=  107  bu.  9  Ib.,  Ans. 

Ex.  22.  50  A.  136.4  P. 

48  «   123.3  « 


200  A.— 99  A.     99.7  P. =100  A.  60.3  P. 
=  100.376875  A.;  $35x100.376875 
=$3513.19+,  Am. 

(211,  212) 


SUBTRACTION.  87 

Ex.  23.  58x37x6:.rl2876cu.ft.=:476cu.yd.24cu.a; 
476  cu.  yd.  24  cu.  ft. 
471      "       16       «     972  cu.  in. 


Ans.   5  cu.  yd.    7  cu.  ft.  756  cu.  in. 
Ex.24.  I  lb.=      9oz.l2pwt.  |  oz.   =12pwt. 

4|oz.=      4"    16    "16gr.      fpwt.=  0    "    21  gr. 
31|pwt.=      1  "    11    «    8gr.      — 


1  Ib.  4  oz.  0  pwt.  0  gr. 
11  «   3  « 

1  Ib,  3  oz.  8  pwt.  21  gr.,  Ans. 

Ex.  25.  5T72  A. =5  A.  2  R.  13  P.  10T^  sq.  yd. 
A.=4  "  0  "  26  "  20£    " 


11  pwt.  3gr, 


R.= 

30  « 

P.= 

18 

a 

9 

A.  3 

R.  30  P. 

18 

sq.  yd. 

4 

«  0 

«   25  " 

12 

ft 

5  A.  3  K.    5  P.    6      sq.  yd.,  Ans. 


(380,  page  213.)       ,, 

Ex.  1.  1783  yr.  1  mo.  20  da.  Ex.  2.  1732  yr.  2  mo.  22  da. 
1775  "  4  "19  "       1620  "12  «  22  " 


s.  7  yr.  9  mo.  1  da.        Ill  yr.  2  mo.,  Ans. 

Ex.  3.  1860  yr.  7  mo.  4  da.  Ex.  4.  1861  yr.  6  mo.  3  da. 
1607  "  5  "  23  "       1859  "  1  "  30  " 


Ans.  253  yr.  1  mo.  11  da.      Ans.  2  yr.  4  mo.  3  da. 
(212,  213) 


88  COMPOUND  NUMBERS. 

Ex.  6      From  July  4,  1855,  to  July  4,  I860,  is  366  da. 

+365  da.+365  da.+365  da.+366  da.=1827  da.; 
From  July  4,  I860,  to  Dec.  12,  1860,  is  27  da. 
+31  da.+30  da.+31  da.+30  da.+12  da.=161 
da.;  from  16  minutes  past  10  o'clock  A.  M.,  to  22 
minutes  before  8  o'clock  p.  M.,  is  9  h.  22  min.; 
hence  1827  da.+161  da.+9  h.  22  min. 
=1988  da.  9  h.  22  min.,  Ans. 

Ex.  7.     1862  yr.  1  mo.    1  da.    4  h.  55  min.  24  sec. 
1860  "  4   "    21   «  12  «  40    "     25    « 

1  yr.  8  mo.    9  da.  16  h.  14  min.  59  sec. 

As  the  full  year  is  a  common  year,  and  the  full  months 
commence  with  May,  the  1  yr.  8  mo.  9  da.=365  da. +31  da. 
+30  da.+31  da.+31  da.+30  da.+31  da.+30  da.+31  da. 
+9  da.=619  da.;  hence,  619  da.  16  h.  14  min.  59  sec.,  Ans. 

Ex.  8.     27—4=23;  242  da.+23  da.=265  da.,  Ans. 


MULTIPLICATION. 

(38»,  page  215.) 

Ex.  3.  Ans.  44  A.  3  E.  2  P.  9  sq.  yd.  6  sq.  ft. 
Ex.  4.  Ans.  131  Cd.  5  cd.  ft.  12  cu.  ft. 

bu.  pk.  qt.  pt.  Ib.  oz.  pwt. 

Ex.  5.  34  3  6  1       Ex.  6.  4  10  18.7 

2  9 


69  3  5  44   2   8.3 

7  3 


489  1  3,  Ana,          132   7   4.9,  An* 
(214,  215) 


MULTIPLICATION.  89 

lb.  3    Z    9     gr.                     gal.  qt.  pt.  gi. 

Ex.  7.         9    3     2    13         Ex.  8.     5    2    1  3.25 

7  12 


5630    11  68    213 

5  8 


27     7     7     2     15,  Am.         549     3  ,  Ans, 

A.    R.    P.    sq.  yd. 
Ex.  9.     78     3     15      15 


1182    2    32      13^  product  by  15 
52     2     10       10        «        «    | 


1235     1       2       23|, 


cu.  yd.  cu.  ft.   .cu.  in. 
Ex.10.       9         10       1424 


75          5      1024  product  by  8; 
9 


676        23         576  product  by  8x9=72 
9        10      1424      «        "  1 


Ans.  686          7         272       "        «  73 

Ex.  11.     27  lb.  2  oz.  17  pwt.  12  gr.;  product  by  10 

163  «  5  "^  5    "      0  «  ,      «        «  10X6=60 
Subtract     2  «  8  «    13    «     18  «  ,       "         «  1 


160  lb.  8  oz.  11  pwt.   6  gr.,       «         «   60—1=59 

Ex.  12.     22  'yd.  0  ft.  11.5  in.,  product  by  5 

111  "    1  "     9.5  "  ,      "        "  5x5=25 
Ans.   557  «    2  «  11.5  «  ,       "        «  5x5x5=125 
(215,  216) 


90  COMPOUND   NUMBERS. 

Ex.  13.  1  qt.  2  gi.=l|  qt.; 

1|  qt.X  144=180  qt.=45  gal.,  Ans. 
Ex.  14.  5  Cong.  5  0. 15  f  I  2  f  3  30  nt,  prod,  by  6 

Ans.   22     "      7  "  13  "    2  «  ,     «      «  6x4=24 

Ex.  15.  14  Hid.  46  gal.  1  qt.  0  pt.  2.4  gi.,  prod,  by  4 

58    "     59   «    1  "  0  "  1.6  "  ,     "     "   4x4=16 
3    «    43   "    0  "   0  "  2.6  "  ,     "    «  1 


S.  62  hhd.  39  gal.  1  qt.  1  pt.    .2  gi.,     «     «  16X1=17 

Ex.  16.  9  T.  13  cwt.  1  qr.  10.5  lb.=9.6671875  T.; 
9.6671875  T.  X  1.7=16.43421875  T. 
=16  T.  8  cwt.  2  qr.  20.65  lb.,  Ans. 

Ex.  17.  2  hhd.  23  gal.  2  qt.  1  pt.^2.375  hlid.j 
2.375  Wid.x4.8=11.4  hhd. 
=11  hhd.  25  gal.  1  pt.  2.4  gi.,  Ans. 

Ex.  18.  9  oz.  13  pwt.  8  gr.xl2=9  lb.  8  oz.=9f  lb.,  whole 

weight;  $212.38 X9§=$2053.00|, 
Ex.  19.  27  gal.  3  qt.  1  pt.=27.875  gal.; 
27.875  gal.x5=139.375  gal.; 
$1.375X139.375  =  $191.64+,  Ans. 

Ex.  20.  37  bu.  3  pk.  5  qt.=37.90625  bu.; 
37.90625  bu.x5=189.53+  bu.; 
$.65X189.53=$123.20— ,  Ans. 


%'at 
DIVISION. 


(3839  page  218.) 
Ex.  6.     Ans.  21  bu.  1  pk.  5  qt.  1  pi. 

yd.    ft.    in. 
Ex.  7.     3)  336    4    3| 

(216-218) 


DIVISION.  91 

7)112     1     5f 

16     0     2|;  Ans. 
Ex.  9.     Ans.  10  cu.  yd.  3  cu.  ft.  428.15  cu.  in. 

mi.     fur.   rd.  yd. 
Ex.10.  9)1986    3     20     1 

12)    220     5     28     5 


18     3       5     4-^,  Ans. 

sq.  mi.    A.     R.      P. 
Ex.11.         12         0       1       30 

2 


45)24         0       3       20 

341       1       16f ,  Ans. 
Ex.  12.  Ans.  1  da.  12|i  h. 

cu.  yd.  cu.  ft.  cu.  in. 

Ex.  13.  33|=-Mp;        3794      20       709f 

3 


Ex.  14.  13|^ 


00)  11384 

7 

400 

113 

22 

1300,  An*. 

Ib. 

I 

3 

9 

gr. 

121 

3 

2 

1 

4 

4 

11)485 

1 

1 

l 

16 

5)44 

1 

1 

1 

16 

3962     S^Ans. 
(218) 


92  COMPOUND   NUMBERS. 

Ex.  15.  28°  51'  27.765';:=28.85771250; 

28.8577125°-r-2.754=:10.4784720 ; 
10.478472°=10°  28'  42£",  Ans. 

yd.     ft.    in.  da.       h.     min. 

Ex  16.  202    1     6|  Ex.  17.  10)  1950    15    15f 

5 


3)  1012     1    9| 


10)195      1    31f 
Ans.  19     12       9 


337     1     7|,  Ans. 
Ex.  18.  4  sq.  mi.-5-124=82  A.  2  R.  12f  f  P.,  Ans. 

Ex.  19.  48^X24X6^=7566  cu.  ft.=280  cu.  yd.  6  cu.  ft.; 

48)280  cu.  yd.  6  cu.  ft. 

5  cu.  yd.  22  cu.  ft.  1080  cu.  in.,  Ans. 

Ex.  20.  2  bu.  3  pk.  6  qt.— 94  qt.;  356  bu.  3  pk.  5  qt. 
=11421  qt.;  11421-^94=121^  Ans. 


LONGITUDE  AND  TIME. 

(385,  page  219.) 

Ex.1,    9h. 

8  "     7  min.    4  sec.      Ex.  2.     1  h.  11  min.  56  sec. 
15 


52  min.  56  sec. 


15  Take   17°     59' 
From  89°       2' 


13°     14'    0"  din7.  Ion. 

77°       V  71 


° 


90°     15',  Ans. 

(218,  219) 


LONGITUDE   AND   TIME.  93 

Ex.  3.     9  h.  13  min.  20  sec.  A.  M.,  time  at  the  easterly  place  ; 
2  h.  30  min.  A.  M.,     "    "        westerly     " 


6  h.  43  min.  20  sec.,  difference  of  time ; 

6  h.  43  min.  20  sec.Xl5=100°  50',  diff.  Ion.; 

100°  50'— 18°  28'=82°  22'  west,  An*. 

Ex.  4.     6  h.  8  min.  28  sec.Xl5=92°  7.',  Ans. 

Ex.  5      1  h.  18  min.  16  sec.  X  15=19°  34'; 

940  44'— 19°  34'=75°  10'  west,  Ans. 

Ex.  6.     2  h.  58  min.  23|  sec.  X  15=44°  35'  50",  diff.  in  Ion. ; 
77°  51'+44°  35'  50"=122°  26'  50"  west,  Ans. 

Ex.  7.     2  h.  33  min.  53i|  sec.  X 15— 38°  28' 29",  diff.  in  Ion.; 
71°  12'  15"+38°  28'  29" 
=109°  40'  44"  west,  Ans. 

Ex.  8.     5  h.  40  min.  20  sec.Xl5=85°  5'  west,  Ans. 

Ex.  9.     12  h.— 5  h.  51  min.  41f  sec.=6  h.  8  min.  18|  sec., 
diff.  time ;.  6  h.  8  min.  18|  sec.  X  15=92°  4'  36", 
diff.  in  Ion.;  92°  4'  36"— 18°  3'  30" 
=74°  V  6"  west,  Ans. 

Ex.  10.  8  h.  53  min.  47       sec. 

3.56sec.X24=          1    "    25.44    " 


•  f  true  time 
Take     8  h.  52  min.  21.56  sec.  (  atN.  York; 

From  10  «    4     «     36.80    "    ship's  time; 

1  h.  12  min.  15.24  sec.,  diff.  time; 
1  h.  12  min.  15.24  sec. X  15=18°  3'  48.6",  Ans. 

(386,  page  221.) 
Ex.  1.     84°  24'— 77°  l'=7°  23',  diff.  in  longitude; 

7°  23'-f-15=29  min.  32  sec.,  diff.  in  time,  Ans. 
(220,  221) 


94  COMPOUND  NUMBERS. 

Ex.  2.     113°  14'— 2°  20'— 110°  54',  diff.  in  longitude; 
110°  54'-^-15=7  h.  23  min.  36  sec.,  An*. 

Ex.  3.     78°  55'+20°  30'=99°  25',  diff.  in  Ion. ; 
990  25'— 15=6  h.  37  min.  40  sec.,  Ans. 

Ex.  4.     74°  1'— 63°  36'=10°  25',  diff.  in  Ion.; 

10°  25'-=-15=41  min.  40  sec.,  diff.  in  time; 
4  h.  30  mm.  p.  M. — 41  min.  40  sec. 
=3  h.  48  min.  20  sec.  p.  M.,  Ans. 

Ex.  5.     71°  7'+5'  2"=71°  12'  2",  diff.  in  Ion.; 

71°  12'  2"--15z= 4  h.  44  min.  48T\.seo.,  diff.  in 
time ;  12  h.  M. — 4  h.  44  min.  48f^  sec. 
=7  h.  15  min.  lljf  sec.  A.  M.,  Ans. 

Ex.6. 

118°+120°=:2380,  diff.  in  Ion.  reckoned  west  from  Pekin ; 
360°— 238° =122°,    «     "  «          «      "Sacramento; 

238°-7-15z=15  h.  52  min.,  Sacramento  earlier  than  Pekin  ;  or 
122°-7-15=:  8  h.    8  min.          «  later       «         " 

Ex.  7.     35°  32'+76°  37'=112°  9',  diff.  in  Ion.; 

112°  9'-f-15=7  h.  28  min.  36  sec.,  diff.  in  time; 
6  h.  40  min.  A.  M.-J-7  h.  28  min.  36  sec. 
—  2  h.  8  min.  36  sec.  p.  M.,  Ans. 

Ex.  8.     6  h.  p.  M.— 7  h.  28  min.  36  sec. 

=10  h.  31  min.  24  sec.  A.  M.,  Ans. 

NOTE. — We  always  subtract  the  difference  of  time  from  the 
time  at  the  easterly  place  to  obtain  the  time  at  the  westerly  place, 
adding  12  h.  to  the  minuend  if  necessary  to  make  the  subtrac- 
tion possible;  and  we  always  add  the  difference  of  time  to  the 
westerly  place  to  obtain  the  time  at  the  easterly  place,  rejecting 
12  h.  when  the  amount  exceeds  this  time.  And  whenever  12  h. 
R,re  borrowed  n  subtracting,  or  rejected  in  adding,  the  time 
changes  from  A.  M.  to  p.  M.,  or  from  p.  M.  to  A.  M. 

(221) 


PKOMISCUOtJS   EXAMPLES.  95 

Ex.  9.    94°  46'  34"— 72°  35'  45"=22°  10'  49",  diff.  in  lon.j 
22°  10'  49"-5-15=l  h.  28  min.  43T%  sec.,  diff  k 
time ;  6  h.  20  min.  A.  M/ — 1  h.  28  min.  43T4^  sec. 
=4  h.  51  min.  16{4  sec.  A.  M.,  Ans. 

Ex.  10.  28°  49'+93°  5'=121°  54',  diff.  in  Ion.; 

121°  54'-=-15=8  h.  7  min.  36  sec.,  diff.  in  time; 

3  h.  p.  M.+ 8  h.  7  min.  36  sec. 

=  11  h.  7  min.  36  sec.  p.  M.,  Ans. 
.Ex.  11.  12  h.  M.— 8  h.  7  min.  36  sec. 

=3  h.  52  min.  24  sec.  p.  M.,  Ans. 

Ex.  12.  88°  1'  29"+5'  2"=88°  6'  31",  diff.  in  Ion.; 

88°  6'  31"-5-15=5  h.  52  min.  26-^  sec.,  diff.  in 
time ;  12  h.  M.+5  h.  52  min.  26^  sec. 
=5  h.  52  min.  26yL  p.  M.,  Ans. 

PROMISCUOUS   EXAMPLES   IK   COMPOUND   NUMBERS. 

(Pag-  222.) 

Ex.  1.     Ans.  55799  gr. 

Ex.  2.     3  cwt.  12  lb.  =  .156  T.;  $15.50 X  .156=82.418,  Aw 

Ex.  3.     27  yd.  2  qr.=110  qr. ;  110  qr.-r-5=22  ells,  Ans. 

Ex.  4.    f>18.945-^-$4.84=:£3.914256 
=£3  18  s.  3  d.  1.6-fqr.,  Ans. 

Ex.  5.     Ans.  130413645  gee. 

Ex.  6.     24  sheetsX8X2=384  pages,  Ans. 

Ex.  7.    1  s.  6  d.=18*d.;  £5  6  s.  6  d.=1278  d.; 

1278-^18=71  yd.,  Ans. 
Ex.  8.     Ans.  37173  1. 
Ex.  9.     Ans.  111111  sq.  yd. 

Ex.  10.  3  gal.  1  qt.  1  pt.=27  pt.;  3  hhd.:=1512  pt.j 
1512  pt.-f-27  pt.=56,  Ans. 

(222) 


96  COMPOUND  NUMBERS. 

Ex.  11.  2£  T.=5000  Ib.;  $.095x5000  =$475,  sold  for; 

$475—  $375.75=$99.25,  Am. 

Ex.  12.  3  Ib.  9  oz.=900  pwt.;  1  oz.  5  pwt.=25  pwt.; 
900-r-25=36=3  &>z.,  Ans. 

Ex.  13.  4  gal.  3  qt.=152  gi.;  2  qt.  1  pt.  2  gi.=22  gi.; 


Ex.  14.  |XT4i  rd.xy=f  yd., 

Ex.  15.  26^X20=530  sq.  ft.=58|  sq.  yd.=58|  yd., 

Ex.  16.  15  T.  3  cwt.  3  qr.  24  Ib.  long  ton  weight=34044 
lb.=  17.022  T.  snort  ton  weight;  hence 
$140xl7.022=$2383.08,  sold  for; 
$  .06  X  34044=  $2042.64,  cost; 

$340.44  gained,  Ans. 

Ex.  17.  (40  ft.-f  36£  ft.)X2=153  ft.,  round  the  room; 

153X22|=3404|  sq.  ft.  in  the  walls; 
36^X40=1460  sq.  ft.  in  the  ceiling; 

3404|  sq.  ft.+1460  sq.  ft.—  1375  sq.  ft.=3489^  sq 
ft.  to  be  paid  for,  at  ^8=2  cents  per  sq.  ft.; 
hence  3489|X2=$69.785,  Ans. 

Ex.  18.  78  Ib.  9  oz.x23=18  cwt.  6  Ib.  15  oz.,  Ans. 
Ex.  19.  33°  2'+30°  41'=63°  43',  Ans. 
Ex.  20.  1  gross  4  doz.=16  doz.  ; 

16  doz.X31=496  doz.,  Ans. 

Ex.  21.  4  sq.  rd.  120  sq.  ft.  84  sq.  in.XlS 
=79  sq.  rd.  264|  sq.  ft.  72  sq.  in.  ; 
160  sq.  rd.—  79  sq.  rd.  262^  sq.  ft.  72  sq.  in. 
=80  sq.  rd.  7  sq.  ft.  72  sq.  in.,  Ans. 

Ex.  22.  $12.025-f-$13  =  .925  T.=1850  Ib.,  Ans. 

Ex.  23.  1  pk.  4  qt.=.375  bu.;  $.72-f-.375=$1.92,  Ans. 

Ex.  24.  36244  lb.-:-60=604TL  bu.,  Ans. 

(222,  223) 


PROMISCUOUS   EXAMPLES.  97 

rix.  25.  32X24X6=4608  cu.  ft.  =  170f  cu.  yd.; 
$.20xl70f  =$34.13+,  Ans. 

Ex.  26.  (32  ft.+24  ft.)X2=112  ft.,'  mason's  girt; 
112X6X1^=1008  cu.  ft.=40T8T  Pch.; 
?,  Ans. 


Ex.  27.  1  mi.=5280  ft.;  10  ft.  4  in.  =  10|  ft.; 

i26720  times,  hind  wheels; 
r_.  84480      «      forward  « 

42240  times, 

Ex.  28.  $3.75X15.22=$57.075 
4.25X  7.36=  31.28 


$88.355  cost; 
1045X2258=  101.61    sold  for; 

$13.255,  Ans. 

Ex.  29.  f  of  3  T.  10  cwt.          =5000  Ib. 
of  7  T.  3  cwt.  26  lb.=4408  Ib. 


592  Ib.,  Ans. 

Ex.  30.  16  Ib.  5  oz.  10  pwt.  13  gr.  Troyr=94813  gr  ; 
94813  gr.^-7000=13.5447+lb.  Avoirdupois; 
13.5447+lb.=13  Ib.  8  oz.  11.4+dr.,  Ans. 

Ex.  31.  1  ft.  7.8  in.  =1.65  ft.  =  .l  rd.,  Ans. 

Ex.  32.  9  in.=|  ft.;  6  in.=|  ft.; 

fX|Xf=8ft.,  Ans. 
Ex.  33.  16  in.=|  ft.;  7X|=5|  ft.,  Ans. 

Ex.  34.  17280  cu.  in.H-268|=:64?  gallons,  dry  measure, 
17280  cu.  in.^-282  =61||      «      beer       « 

Difference  ^s|-g  gallons, 

(223,  224) 


98  COMPOUND   NUMBERS. 

Ex.  35.  6T°Xf  XT!  ft.X51o=2-275  tons,  Ans. 

Ex.  36.  7^ 

—  bu.  =  3  pk.  44  qt. ;  §X%§X  3    q^*-— If  q^*j 


3pk.4f  qt.+lfqt.=3pk.6lfqt; 
5  bu.  3|i  qt.— 3  pk.  6|f  qt. 
=4  bu.  5|  qt.^=16|  pk.,  ^.TIS. 

sq.  mi.    A.        R.      P.  sq.  yd.  mi.        A.         R.        P.     sq.  yd, 

Ex.37.       5    250     3     0    0  456    3     14    25 

7  •  o^ 


8)37    475     1  2      90     2      4     14 
114    0    33     21 


4    459     1    25  

2     204     2    38      5|  2     204     2     38 


2     254     2    26    24|,  Ans. 

Ex.  38.  14.2878X5.6=80.01168  Ib. 

=  80  Ib.  2  pwt.  19.2768  gr.,  Ans. 

Ex.  39.  |  of  24=15  carats  fine,  Ans. 

Ex.  40.  24—16=8  carats=^=|  alloy,  Ans. 

Ex.  41.  384|  A.=384  A.  3  R.    8  P. 

22  "   1  "   20  " 


2)  362  A.  1  R.  28  P.    (See  Prob.  33,  p.  64.) 

181  A.  0  R.  34  P.,  younger ; 
203  «   2  u   14  " ,  elder. 

Ex.  42.  4000  bu.Xf §=3586^  bu.,  Ans. 

Ex.  43.  110  bu.x64=7040  Ib.  clover  seed  exchanged; 

7040  lb.-4-60=117|  bu.  clover  seed,  N.  Y.  measure; 

1171  Xf  X|  —  704  bu.  corn  received,  N.  Y.  measure; 

704 X5T8X 5^=7294  bu.  corn,  N.  J.  measure; 
(224) 


PROMISCUOUS   EXAMPLES. 


8f  X7294=$48632T,  corn  brought  in  N.  J.; 
$4X^10  =$440,  clover  seed  worth  in  N.  J. ; 


$  46^,  the  N.  J.  farmer  gained ; 
HOxfXf— 660  bu.  corn,  he  would  have  received 
had  the  standards  of  measure  been  the  same ; 
hence  729$  bu.— 660  bu.=694  bu.  corn,  gained  by 
the  N.  J.  farmer  in  the  reckoning. 

Ex.  44.  763.4X763.4=582779.56  sq.  ft. 

=13  A.  1  R.  20  P.  164.56  sq.  ft.,  Arts. 

Ex.  45.  20^  ft.X2=41  ft.,  width  of  both  sides; 
42X41=1722  sq.  ft.=17.22  squares; 
$4.62|X17.22=$79.64|,  An*. 

Ex.  46.  17  T.  15  cwt.  62'  lb.=17.78125  T.; 

$1333.593--17.78125=$75,  nearly,  Ans. 


Ex.  47. 


4 

3 

231 


4 
15 
5 
1728 


Ex.  48. 


96 


300 
112 


$350,  Ans. 


77  I  14400 


•>  Ans- 


Ex.  49. 


72 


300 
112 


Ex.  50.        I  300 
90  I  112 


3     1400 


3     1120 


.66|,  Ans. 


$373.33^,  Ans. 


Ex.  51. 


56 


300 
112 


Ex.  52. 


60 


300 
112 


,  Ans. 


8560,  Ans 


(224,  225) 


100  COMPOUND   NUMBERS. 


Ex. 

RO      i  g  \/  <R  6  4  

L)O.         ?f      /{fyrj^}—^ 

$5.777 

¥x  41= 

7.50 

yx  4S= 

2.177 

fx  Vk= 

1.125 

SX  41= 

1.0625 

VX  41= 

4.125 

$21.76+, 

Ex.  54.  (22Xl30)-~-110=26, 
.Ex.  55.  1000X77^8  X^?J-=1334f  oz.=8  Ib.  5||  oz., 

EX.   56.    |g|44X^44_±4|56==28TV_, 

Ex.57.  SXlXf 


Ex.  58.  18  ft.  9  in.=225  in.;  225  in.x3=675  in.  per  sec; 
J^L|«  1^^=14080  sec.  =3  h.  54  min.  40  sec.,  Ans. 

Ex.  59.  He  had  come  from  the  East,  because  his  watch  showed 
later  time  ; 
and  1  h.  6  min.  52  sec.  X  15=16°  43',  Ans. 


Ex.  61.  3  hhd.  9  gal.  3  qt.=:198.75  wine  gal.  ; 
198.75X231^45911.25  cu.  in.; 
45911.25--277.274—  165.5807+Imperial  gal.,  Ans 

Ex.  62.  105.85  ch.x40.15  ch.  =4249.8775  sq.  ch. 
=424.98775  A.,  Ans. 

Ex.  63.  Since  A  has  T\  of  the  farm,  B  must  own  T63  ;  and 
•£%  —  T63=  |,  the  difference  between  their  shares,  which 
is  15  A.  1  K.  28!  R  Hence  I5  A-  l  R-  282  p- 
X6=92  A.  2  E.  11  P.,  whole  farm;  92  A.  2  R. 
11  P.-j-12=7  A.  2  R.  34|  P.,  T^  of  farm;  7  A. 
2  R.  34|  P  X5=38  A.  2  R.  11|  P.,  B's  share,  Ans, 
(225,  226) 


PROMISCUOUS   EXAMPLES. 


101 


41x40xTt5o=16-40  squares  in  both  sides; 
83.40X16.40=855.76,  An*. 

189.5X150=28425  sq.  rd.=177.65625  A.; 
$31.75X177.65625=$5640.58+,  Ans. 

9.75  tons  X  50  =487.5  cu.  ft.;  and  since  every  cubic 
foot  will  make  12  square  feet  of  inch  stuff, 
487.5  cu.  ft.  X  12  =  5850  sq.  ft.  boards,  Ans. 

'  gal-, 


Ex.  64. 
Ex.  65. 
Ex.  66. 


Ex.  67. 
Ex.  68. 
Ex.  69. 

Ex.  70.  Since  the  lots  are  to  be  square,  the  side  of  one  must 
be  a  common  measure  of  the  two  dimensions  of  the 
land.  The  greatest  common  divisor  of  201|  rods 
and  41|  rods  is  f  f  rods,  which  is  the  length  of  one 
side  of  the  required  lots.  Now,  if  we  divide  the 
piece  of  land,  by  running  lines  both  crosswise  and 
lengthwise,  at  a  distance  asunder  of  f  f  rods,  we  shall 
have  41|-f-ff =9  ranges  of  lots,  and  201f-^-f  f  =44 
lots  in  each  range.  Hence  44x9=396  lots,  Ans. 

Ex.  71.  The  quantity  of  land  in  each  of  the  equal  lots  must 
be  the  greatest  common  divisor  of  the  quantities  in 
the  several  pieces.     Hence, 
4A.3  R.  20  P.=  780  P.      4    780  ..  1092  ..  1560  ..  1872 


6  "  3  "  12  "  =1092  "       3 
9  "  3  «  =1560  «      13 

11  «  2  «  32  «  =1872  " 


195  ..    273  ..    390  ..    468 

65  ..      91  ..    130  ..    156 

5  ..        7  ..      10  ..      12 


4X3X13=156  P.,  Ans. 


(226) 


102  DUODECIMALS. 

DUODECIMALS 

ADDITION  AND   SUBTRACTION. 

(388,  page  227.) 

Ex.4.     50  sq.  yd.  1  sq.  ft.  7'  4" 

62      "       0      "      5'  3" 
48      «      2 

42      "      2      «     3'  4" 


202  sq.  yd.  6  sq.  ft.  3'  11",  area  of  the 
300  sq.  yd.  2  sq.  ft.  5'        ,  whole  area ; 

97  sq.  yd.  5  sq.  ft.  1'    1",  An*. 


MULTIPLICATION. 

(39O,  page  228.) 

Ex.1      18ft.         5'  Ex.2.    23ft.         1' 

16ft.         8'  3  «          8' 


12  3' 4"  15  8' 8" 

294  8'  70  9' 


806  sq.  ft.  11'  4",  Ans.  86  sq.  ft.    5'  8" 

4ft. 


345  cu.  ft.  10'  8"= 
2  Cd.  5  cd.  ft.  9  cu.  ft.  10'  8",  Ans 

(227,  228^ 


MULTIPLICATION.  103 

Ex.  3.          79  ft.        8'  Ex.  4.  7  ft.        6' 

38  «      11'  3  «         3' 


8  sq.  yd.    1  sq.  ft.  0'  4"  1  sq.ft.  10'  6" 

336       «        3  4'  22      «      6' 


344  sq.  yd.    4  sq.  ft.  4'  4",  An*.  24  sq.  ft.  4'  6" 

1  ft.       10' 


20  cu.  ft.  3'  9" 
24      «      4'  6" 

44  cu.  ft.  8'  3,", 

Ex.  5.     56  ft.X7=392  ft.,  entire  length;  392 
=43120"=299  cu.  ft.  5'  4",  Ans. 

Ex.  6.        60  ft.       6'  24  ft.        2' 

40  «        3'  •  2 


100ft.       9'  48ft.       4' 

2  60  "        6' 


201  ft.       6'  24  ft. 

22  "  2895  " 


4433  sq.  ft.,      in  walls ;       2924  sq.  ft.  2' 
2924      "     2'    "roof; 
523      «    3'    "gables; 

7880  sq.  ft.  5',  Ans. 

(229) 


104  DUODECIMALS. 

CONTRACTED  METHOD. 

(391,  page  229.) 

Ex.1.      7ft.  3' 4"  5"'  Ex.2.  36  ft.  9'  4" 

6"  8'  5ft.  9"  6"  26  ft. 


36     5'  106  sq.  yd.  2  ft.  3' 

4  10'  2      «       0  «  5' 

4/  2  "  3' 


41  ft.  7'd=,  ^TW.  Ans.  108  sq.  yd.  4  ft.  ll'rh 

Ex.  3,         6  ft.    2'  7"  Ex.  4.          7  ft.  6'  8" 

4"  3'         3  ft.  11"  2'  3  ft. 


18          8'  22      8' 

17'  13' 

2>  r 


20  ft.     5'db  24  ft.  6'± 

6"   8'         2ft.  4"  8'      3ft. 


40        10'  73      6' 

13          7'  16      4' 

10'  8' 


55  ou.  ft.  3'zb;  Am.  90  ft.  6'±,  An*. 

(229,  230) 


DIVISION.  106 

DIVISION. 


(392,  page  230.) 

Ex.  1.  17  ft.)  287  ft.  T  (16  ft.  11',  An*. 
17 

117 
102 

15  ft.  7' 
15  «  1' 


Ex.  2.  6  ft.  8')  29  ft.  5'  4"  (4  ft.  5',  Ans. 

26  «  8' 

2  ft.  9'  4" 
2  "  9'  4" 


Ex.  3.  48  ft.  6')  1176  ft.  1'  6"  (24  ft.  3',  An*. 
1164  « 


12  ft.  V  6" 
12  "  1'  6" 

Ex.4. 

38  ft.  10'   362  ft.  5'  4")275  cu.  yd.  5  cu.  ft.  1'  4"= 
9ft.  4'   362  ft.  5'  4")7430  Cu.  ft.  1'  4"(20  ft.  6',  Ans. 

7248  cu.  ft.  10'  8" 

12   11"  4"         

349   6'  181cu.ft.  2' 8' 

181cu.ft.  2'  8" 

362ft.  5'  4"          


(230,  231) 


106  DUODECIMALS. 

CONTRACTED  METHOD. 

(393,  page  231.) 

Ex.  1.  2  ft.  10'  7")  7  ft.    7'    3"  (2  ft.  7'  8"±,  Ans. 
5  «    9'    2" 


1  ft.  10'    V 
1ft.    8'    2" 

i'  ii" 
i'  ii" 

Ex.2. 

7  ft.  2'  4"  33  ft.  5'  6")  64  ft.  9'  8"  (1  ft.  11'  3"±, 
8'"  9"  7'  4  ft.  33  «  5'  6" 


28    9'  4" 

31  ft.  4'  2" 

4    2'  4" 

30  «  8'  1" 

5'  &" 

5" 

8'1" 

8'  4" 

33  ft.  5'  6"± 


Ex,  3.  7  ft.  2'  11")  36  ft.  4'  8"  (5  ft.  3"  T"±,An$. 
86  ft.  2'  7" 


4" 
4" 

(231) 


SUBTRACTION. 


107 


Ex.  1.  1000 
756 


SHOET  METHODS. 

FOR   SUBTRACTION. 

(395,  page  232.) 

Ex.  2.  4000000 
8576 


244,  Ans. 

Ex.  3.  10.0000 

.5768 


9.4232,  Ans. 

Ex.  5.  64000.00000 
90.59876 


63909.40124,  Ans. 

Ex.  7.  1000    100000 
271     18365 


3991424,  Ans. 

Ex.  4.  1700000 
13057 


1686943,  Ans. 

Ex.  6.  1000000 
599948 


400052,  Ans, 


10000000 
3401250 


729;    81635;    6598750,  Ans. 


Ex.  1.  78400 

784 


FOR    MULTIPLICATION. 

(396,  page  233.) 

Ex.  2.  5873.000 
5873 


77616,  Ans. 

Ex.  3.  478300000 

4783 


5867.127,  Ans. 

Ex.  4.  75000.000 

75 


478295217,  Ans. 


(232,  233) 


74999.925,  Ans. 


108  SHORT   METHODS 

(397,  page  233.) 

Ex.  1.  78600  Ex.  2.  432700 

1572  17308 


77028,  An*.  415392,  Ans 

Ex.3.  7328000          Ex.4.  7873586000 
21984  39367930 


7306016,  Ans.  78342.18070,  Ans. 

Ex.5.  437890000       Ex.6.  7077364000000 
262734  49541548 


437627266,  Ans.        7077314.458452,  Ans. 

(398,  page  234.) 
Ex.1.   567X13  Ex.2.   439603x10.5 


7371,  Ans.  4615831.5,  Ans. 

Ex.3.    7859X107        Ex.4.    18075x1008 


840913,  Ans.  18219600,  Ans. 

Ex.  5.     3907X10.002 


39077.814,  Ans. 

(399,  page  235.) 
Ex.  1.        56783x71  Ex.  2.          47.89x60.1 


4031593,  Ans.  2878.189,  Ans. 

Ex.  3.          3724.5X.901         Ex.4.  103078x40001 


3355.7745,  Ans.  4123223078,  Ans. 

(233-235) 


FOB  MULTIPLICATION. 


109 


Ex.1. 


Ex.3. 


Ex.  1 
Ex.  2 

Ex.3 

282= 
392= 
372= 


(4OO,  page  236.) 

432711000  Ex.  2. 

432711 


9)432278289 

48030921 
2 

96061842,  Am. 

673200000 
6732 


Ex.4. 


9)673193268 

74799252 

8 

59.8394016,  Ans. 

Ex.5.    4444400000 
444.4.4. 

J.  J-  JL  J.  X 

9)4444355556 

493817284 
8 

3950538272,  Ans. 

(401,  page  237.) 
24xBO+32=729,  Ans. 
48X50+12=2401,  Ans. 


5780000 
578 

9)5779422 

642158,  Ans. 


8675000 
8675 

9)8666325 


674047.5,  A** 


=  784  26*=22x30+42  =  676 

38X40+F=1521  382  =36x40+2*  =1444 

34X40+32=1369  362=32x40+42=1296 

352=30X40+52=1225. 

(236,  237) 
10 


110  SHORT   METHODS 

Ex.4. 

77*:=  74X80+3*  =5929;  882=86x90+22=7744  ; 
8.62  =  8.2x9.0+.42=73.96  ; 

99*  =98x100+1*  =9801;  982=96xlOO+22=9604  ; 

692=68X  70+12=4761;  682=66X  70+22=4624^ 

6.72=6.4x7.0+.32=44.89;  62*=60x  64+22=3844. 


(4O3,  page  237.) 

Ex.  1.  43700--4=10925,  Ans. 

Ex.  2.  68720-f-4=17180,  Ans. 

Ex.  3.  5734154000--3=:1911384666f;  Ans, 

Ex.  4.  75864200-r-8=9483025?  Ans. 

Ex.  5.  78563000--8  =9820375,  Ans. 

Ex.  6.  57687000-f-7=8241000,  Ans. 

(4O4,  page  238.) 

Ex.  1.  43789X100X81=36125925,  Ans. 

Ex.  2.  58730X1000X7^=418451250,  Ans. 

Ex.  3.  7854X34|=268999.5,  Ans. 

Ex.  4.  30724X10000X7^=22530933331,  Ans. 

Ex.  5.  47836X100X7^=34083150,  Ans. 

Ex.  6.  53727X100X24^=129840250,  Ans. 

(4O5,  page  239.) 
Ex.  1.     $568-M=$142,  Ans. 

Ex.  2.     $51-^-6=$8.50;  33x8=264  yd.;  $264^-16 
$16.50;  $18-f-3=86;  $8.50+$16.50=$25; 
$25-46=$19,f  Am 

Ex.  3.     $28-f-8~  $3.50,  Ans. 

Ex.4.     $576-4-9=$64,  Ans.  • 

Ex.  5.     $7.875-f-63=8.125=$£,  gain  on  1  gal.j 
$576-r-8=$72,  Ans. 

(237-239) 


FOR  MULTIPLICATION.  Ill 

(4O6,  page  241.) 


Ex.  2.      $15.46 
46 


$711.16,  cost  of  3  Ib.  10  oz.=46  oz.j 
.773    «    «  lpwt.=^of  loz.; 
5.411    "    «  7    " 
.129    «    "  4gr.=  i  oflpwt.; 
.032    "    "   1   " 
.016.    "    "  i  " 


$717.521, 
Ex.  3.    90  lb.=.9  cwt.;  $.56x5.9=$3.304;  Art*. 

Ex.4.    $4.48 
3 


$13.44,  cost  of  3  bu.; 

1.12    "    "  lpk.=jbu.; 
.14    «     «   lqt.=4pk.; 

28    "    "  2   " 

$14.98,  An*. 

Ex.  5.      8  Ib.    5  oz.      6.74  dr. 

5 


41  Ib.  11  oz.     1.7       dr.,  weight  of  5  gal. ; 
2   "      1    "     5.685     «        «      "  1  qt.r^|  ^1, . 

4  "      2    «  11.37      "        «      "  2  « 

1  "  0  «  10.842+"  "  "  lpt.=^qt.; 
4  "  2.71  "  "  "  lgi-=i  PM 
8  «  5.42  «  «  «  2  " 


49  Ib.  12  oz.    5.73— dr., 
(241) 


112  SHOKT   METHODS 

Ex.6.    $17.50 
3 


-    $52.50  ,  cost  of  3  A.; 

4.375,  "    "    IE.; 

.547,  "    "    5  P=£  K.; 

3.282,  "    "  30  P.; 

•043,  "    "   .4P.=T^ 

$60.747—,  ^TIS. 
Ex.  7.      £3  17  s.  10.5  d. 

;..  7 


£27     5s.     1.5    d.,       val.  of  7  oz.; 

1  18  "  11.25  "        "    «  10  pwt=,J  oz.; 
19"     5.625"      "    «    5    " 
3  "  10.725  "       "    "     1    " 
1"  11.3625"       "    «     12gr.=^pwt.; 
11.68125"    "    "    6     "=|of!2gr.; 

£30  10  s.    4.14375  d.,  Ans. 


Ex.  8       4°  36'  40" 
5 


23°     3'  20",      in    5  da.; 
2°  18'  20",       "  12h.=  i  da.; 
34'  35",       "     3  " 
5'  45|",     "  30  min.=J  of  3  h.; 
23T^",  "     2     " 
5f|",   "  30sec.=*  of  2min.; 
"  15   « 
"  10   « 


26°     2'  34|||",  Ans. 
(24X) 


FOE   DIVISION.  1  1  3 

Ex.  9.       7  gal.  1  qt.  1  pt.  3  gi.=7.46875  gal.; 
hence,  $7.46875,        cost  at  8  s.; 

$3.734375,        «     "4s.; 
.622395+,     "     «  8  i; 

$4.35+,  Ans. 
Ex.  10.  $12.50  ,    at  6  s.  per  day; 

£  2.083+,  «  1  s.   «     " 


$10.416  +,  "  5  s.   "    " 
.521—,  «  3  d.   "    « 


$10.93+, 


FOR  DIVISION. 

(4O7,  page  242.) 

Ex.  1.  634.75X4=2539,  Ans. 

Ex.  2.  785.6X8=6284.8,  J.TIS. 

Ex.  3.  5.16X3=15.48,  Ans. 

Ex.  4.  .167324X8=1.338592,  ^TW. 

Ex.  5.  1748X7=12236,  Ans. 

Ex.  6.  57.634X6=345.804,  Ans. 

(4O8,  page  248.) 

Ex.  1.        2575  )  64375 
4  4 


103/00  ). 2575/00(25, 
206 


515 

515 

(241-243) 


114  SHORT  METHODS. 

Ex.  2.        3625  )  76394 


29/000)  611/152  (2l$$s,  Am. 

58 

31 
29 


2152-5-8=269 


Ex.  3.    4331  )  7325 
3  3 


13/00  )  219/75  (16f  |, 
13 


78 

*»  _ 

1175x4=47/00 
1300x4=52/00 

Ex.4.      431.25  )  5736 


3450         45888 
2  2 

69/00      )  917/76  (IStff,  Ant. 
69 

227 

207 

2076—12=173 

(243) 


RATIO.  115 


Ex.  5.     566f  )  42.75 
3  3 


17/00)  128/.25  (j07|f,<£ifc 
119 

9.25X4=37 
17.00X4=68 

Ex.6.   .21875  )  24409375 


1.75    195275000 
4          4 


)781100000 


111585714f ,  AM. 


Ex.7.    3.14f)    785 


22  4095 


249^,  Ans. 


RATIO. 

(481,  page  246.) 
Ex.  2.     ;&=$,  Ans.  Ex.  3.     T82%=|,  Ans. 

60 
Ex.  4.    — =7,  Ans.  Ex.  5.     ^X  V=48f, 


Ex.  6.  2$-s-7i=f  X  &= 
Ex.  7,  A-s-J=AXf =4i, 
Ex.  8,  1  mi. =8  fur  ;  |, 

(243—246) 


116  PROPORTION. 

Ex.  9.     1  wk.  3  da.  12  h.=252  h.;  9  wk.=1512  h.j 

1512-=-252=6,  An*. 
Ex.  10.  10  A.  1  R.  20  P.  =1660  P.; 

6  A.  2  E.  30  P.  =  1070  P.; 

1070-5-1660={gJ,  An*.* 
Ex.  11.  25  bu.  2  pk.  6  qt.=822  qt.; 

40  bu.  4.5  pk.=1316  qt.  ;  1316--822=lf  ff,  An* 

Ex.12.  18f°=67500";  45'  30"=2730"; 


12  * 


Ex.i4.  | 

Ex.  15.  42-*-28=l£,  ^TW.     (See  415.) 

Ex.  16.  43  gal.=172  qt.; 

Ex.  17.  15X|=12, 

Ex.  18.  3|-5-7=^|, 

Ex.19.  .75- 

Ex.  20.  $6.125-f-25=$.245,  An*. 

Ex.21.  |XS=A,^«. 

Ex.  22.  13  A.  3  B.  25  P.X|§=6  A.  2  B.  10  P.,  An*. 


PBOPOBTION. 

(428,  page  248.) 

26X10 

Ex.  1.     (?)= =65,  An*. 

4 

16  A.X8865 

Ex.  2.     (?)= =197  A.,  An*. 

720 

(246-248) 


SIMPLE   PROPORTION.  117 

4^  yd.X29.25 

Ex.  3.  (?)=  --  =134  yd.,  Ans. 
•  9.75 

Ex.  4.  21  A.  3  E.  20  P.  =3500  P.; 

3500  P.X1260 

(?)=  -  =5880  P.=36  A.  3  R.,  Am. 
750 


Ex.  5.     (?)=  V%6  oz.X^XT15=2il  oz.,  Ans. 
Ex.  6.    £407  2s.  lOf  d.=£4074=£2-8^  J 

(?)=7  oz.X2-875-J)X31o=95  oz.=7  Ib.  11  oz.,  Ans. 

.15  hhd.x2.39 

Ex.  7.     (?)=  -  =1  hhd.,  Ans. 
.3585 

Ex.  8.     1  T.  7  cwt.  3  qr.  20  lb.=3128  Ib.; 
13  T.  5  cwt.  2  qr.=29736  Ib.; 


Ex.  9.     (?)=$175.35xf  Xf  =$601.20,  Ans. 
Ex.10.  $12i=$\6;  2404=1-67^1 

(?)=«V  X  H8-1  Xjiif  ?u= 
Ex.  11.  |  yd.X6Tm¥Xf  =40^  yd., 


.SIMPLE  PROPORTION. 


(436,  page  252.) 
Ex.  1.  12  gal.:  63  gal.=$30  :  (?) 


Ex.  2.    9  bu. :  100  bu.=2  bar. :  (?) 

2  bar.  X 100 

(?)= =22§  bar.,  Ans, 

9 

(248-252) 


1 1  8  PKOPOKTION. 

Ex.  3.     $26.75— $22.25— $4.50,  gain  on  18  bu.; 

18  bu.  :  240  bu.=$4.50  :  (?)  • 

(?)=$4^|^-°=$60,  Ans. 

Ex.4.     6£bu.  :9|  =  $3  :  (?) 

(?)=$3X347+f3  =  $4-269+,  Ans. 

Ex.  5.     1|  yd.  :  87^  yd.=$.42  :  (?) 

(?)=$.42X-i-FX4=$21,  Ans. 

Ex.  6.     $1500X2V0i?40o:=^5446-62+?  Ans- 

Ex.  7.  |  of  15  da. =12  da.;  and  since  the  number  of  men 
required  will  vary  inversely  as  the  time,  20  men 
Xyj|  =  25  men  required  to  perform  the  work  in  12 
days;  25  men — 20  men =5  men  to  be  added. 

Ex.  8.     100  yd.  :  3.25  yd. =$473.07/3  :  (?) 
^i5nxi_3XT£_=$i5.375;  Ans. 

Ex.  9.     1  Ib.  4  oz.  10  pwt.=330  pwt.; 

330  pwt. :  (?) =$260.70  :  $39.50 

330  pwt.  X 260.70 

(?)=— =2  oz.  10  pwt,  Ans. 

39.50 

Ex.  10,  1  h.  14  min.Xtf =2  h-  46  min-  30  sec->  *Ans- 
Ex.  11.  $if:83V=*bn.:(?); 

(?)  =  £  bu.X570Xfi=T73z  bn.,  Ans. 
Ex.  12.  46  A.  3  K.  14  P. =7494  P. ;  35  A.  2  E.  10  P. 

=  5690  P.;  $374.70xf!M=$284.50,  Ans. 

Ex.  13.  1  yr.  3  mo.=15  mo.;  2  yr.  8  mo.=32  mo.; 
$1870.65Xff =$3990.72,  Ans. 

H 

Ex.  14.  164.5X—=246.75;  Ans. 
5 

Ex.  15.  12  A.  3  R.  36  P.=2076  P.; 

2076  P.x  V^=17127  P.=107  A  7  P.,  Ans. 
(252,  253) 


COMPOUND   PKOPORTION. 


119 


Ex.  16.  $325  :  $2275=$26.32  :  (?); 

$26.32X2275 

(?)= =$184.24,  Ans. 

325 

Ex.  17  60.5X44=2662  sq.  ft.=295|  sq.  yd. ; 
14|  sq.  yd.  :  295|  sq.  yd.=$34i  :  (?); 
(?)=$JLp-X2-V- XA=$709.86|,  Ans. 

Ex.  18.  1  doz.  =  12;  lOf  gross=1548; 
§.0625xif|-a=$8.06|,  Ans. 

Ex.  19.  7  s.  6  d.  =  7^s.=  136  s.;  and  since  the  weight  of  the 
loaf  should  vary  inversely  as  the  price  of  wheat,  we 
have  7£  s.  :  6s.  :=(?)  :9  oz.j 

(?)=9  oz.XV5X^=Hi  oz.,  Ans. 


Ex.1. 


Ex.2. 


COMPOUND  PKOPORTION. 

(4395  page  256.) 


(12    ((?) 

:   J    =  11  : 
I  5    (18 


33 


Ex.3. 


12 


16  16 

=  1260  :  4728   (?) 

(?)  1260 


(253-256) 


18 
11 


12 

5 

33 


(f)=10,  Ans, 


12 

4728 


10  |  197 
(?)=19.7  da.,  Ans. 


120 


PROPORTION. 


Ex.4. 


144 

6 

12 


30  (  200 

(?)=]  3 
7  (2 


350 

6 

3 


144X6X12X350X6X3 

(?)= — =259.2  da.,  An*. 

30XTX200X3X2 

Ex.  5.     5  bu.=20  pk.;  3  bu.  3  pk.=15  pk.; 

22  da.XlXff^44  da->  Am. 


Ex.  6. 


10|  . 

4 


3000 


=1:1 


(0 

32 
3 
3 
4 


3000 

3 

38 

2 

11 


Ex.  7.      T300 


("300        ("(?)  (?) 

\          :    ]        =  1  :  8     .90 
(1.25      (.90  1 


300 
1.25 
3 


Ex.  8.        468 


(?)=1250  bu.,  An*. 
[120 
6 


468X120X6 
(f)= =3240,  An*. 


26X4 


Ex.  9. 


io 

l3 


i  (16 

*  :       17      =546  : 
(15 


Ex.  10.  7 


=384  bbl,  Ant. 
{^=22  da.,  Ant. 
(256) 


PROMISCUOUS   EXAMPLES.  121 

PROMISCUOUS  EXAMPLES  IN  PROPORTION. 

(Page  257.) 

Ex.  1.     7  ft.  :  198  ft. =4  ft.  :  (?) 
4  ft.  X 198 

(?)  = =1134  ft.,  Ans. 

7 
Ex.  2.    $972  :  $11|=$607^  :  (?) 

Ex.  3.     3  cwt.  X  Vo0  X  ¥ =99  cwt.,  Ans. 
Ex.  4.     18  da.XT%=14£  da.,  Ans. 
Ex.  5.      ( (?)  (  140 

(16.50   *      (24.75 

140  A.X24.75 

(?)  =  _ -210  A.,  Ans. 

16.50 

Ex.  6.      (1728          (750 

4  :     1  =$155.52  :  (?) 

(18          (54 

$155.52X750X54 

(?)= =$202.50,  Ans. 

1728X18 


Ex.  7. 

28  oz 


(  22          ( 
.  :  (?)=  J          :     4 
(6  ( 


Ex.  8.     15  menX25°=60  men,  Ans. 
Ex.  9.     12s.  7d.=151d.;  1  oz.=240  gr.; 

15  Ib.  11  oz.  13  pwt.  17  gr.=92009  gr.; 

240  gr.:  92009  gr.=151d.:  (?) 

151d.x92009 

(?)=  -  =28944.497+d.  ; 
240 

28944.497d.-7-Q6=$301.50+,  Ans. 

(257) 
11 


122 

PROPORTION. 

Ex.  10. 

(16 

(171 

(?) 

$36.72 

1    7    : 

\  10i=$36.72  :  ( 

0      16 

17.5 

(15 

(16* 

7 

10.5 

15 

16 

(?)=$64.26,  Ana. 

Ex.  11.  2  yr.  5  mo.  5  da.=29|  mo.;  1  yr.  8  mo. =20  mo.; 
and  since  the  money  lent  should  vary  inversely  as 
the  time,  we  have  291  mo.  :  20  mo.=(?)  :  $1200; 
$1200X291 


v 

.'•J  — 

20 

:tjpjLi«ju,   jtins. 

Ex. 

12.  < 

59  :  f 

?5=$10.50  : 

$5.83|;  Ans. 

Ex. 

13. 

(12 

1    9 

(CO 

:     \    7=2 

:  I 

(?) 

7 

12 

9 

15f 

(15 

15 

2 

9     140 


12  —  8=4  men,  Ans. 


(?)  —  8 


Ex.  14.  1  yr.  8  mo.=20  mo.;  3  yr.  4  mo.  24  da.=40.8  mo.; 

'300  210.25 

=$30  :  (?) 
40.8 

$30X210.25X40.8 
(?)=  ---  =$42.891,  Ans. 


Ex.  15.    (  (?) 


300X20 
2     =1  :  1 


Or, 


4 
19yd. 


(?)=y>  yd.X|X|-15|  yd,  Ans.    (  )=15|  rd.,  Ans, 

Ex.  16.    (24          (38 

=$95.60  :  (?) 


18 


22 


;  Ans. 


(257,  258 


PROMISCUOUS   EXAMPLES.  123 

Ex.  17.  16^  Cd.  :  0)=11A  T.  :  15£  T. 

0)  =  W  Cd.xWX3%=22|  Cd.,  Am. 

Ex.18.       8  18 

s=75  :  450 

0) 


?V  =  7I&   ft'>  AUS- 

Ex.  19.    (  4  (  15 


0)=V  A. 

Ex.  20.  600  XT%  XVs  =450  men,  Ans. 

Ex.  21.  120  gal.—  80  gal.=40  gal.  gained  in  1  hour; 
20  bar.  =630  gal.  to  be  be  filled; 
40  gal.  :  630  gal.=l  h.  :  15|  h.,  Ans. 

Ex.  22.  16  oz.  —  14T9g  oz.=lT7s  oz.=f  |  oz.      Now,  if  every 
16  ounces  of  groceries  have  been  lessened  f  |  ounces, 

how  much  should  the  price,  838.40,  be  lessened  ? 
16  oz.:  1/5  oz.=$38.40  :  $3.45,  Ans. 

Another  Solution. 

If  every  16  oz.  be  reduced  to  14T9g,  to  what  sum  should 
the  cost,  $38.40,  be  reduced  ? 

16  oz.  :  14T90=$38.40   :  $34.95 
$38.40—  $34.95=$3.45,  Ans. 

Ex.  23.  75  Ib.  coffeeXf=120  Ib.  sugar; 
5  Ib.  :  120  lb.=$.625  :  $15,  ^7^. 

Ex.  24.  $28.75X3|§u=$35.38+,  Ans. 

Ex.  25.    (  12000          (  3000 

J8  :      -|l2     =859|:(1) 

(  550  (  320 

=83     Am 


(258) 


PERCENTAGE. 
'24       f   7 


PERCENTAGE. 

NOTATION. 


(442,  page  260.) 

Ex  1.  .03  ;  .09  ;  .12  ;  .16  ;  .23  ;  .37  ;  .75  ;  1.25  ; 
1.84;  2.05. 

Ex.  2.  .15  ;  .11 ;  .045  ;  .0525  ;  .0875  ;  .205  ;  .25625  ; 
.356;  .24875;  1.305. 

Ex.  3.  .0025  ;  .0075  ;  .005  ;  .004 ;  .00375 ;  .0028  ; 
.00288;  .013125;  .102. 

^.4.  2*;  f;  iVs;  i;  f;  TT;  if>^- 

Ex.  6.     .065 =.06^=6^  per  cent.,  Ans. 
Ex.  7.     .14375=.14|=14|  per  cent.,  Ans. 
Ex.  8.     .0975  =  .09|=9|  per  cent.,  Ans. 
Ex.  9.     .014=.01f =1|  per  cent.,  Ans. 
Ex.  10.  .1025=.10|=10^  per  cent.,  Ans. 
Ex.  11.  .004=.00|=f  per  cent.,  Ans. 
Ex.  12.  .028=.02|=2|  per  cent.,  Ans. 
Ex.  13.  .1324=.13^=13565  %,  Ans. 
Ex/ 14.  .084f=.083-83  %,  Ans. 
Ex.  15.  ,004T6T=.00T\=f\  %,  Ans. 
Ex.  16.  .003T^=.00T43=T4^  %,  Ans. 

(258—260) 


GENERAL  PROBLEMS.  125 

GENERAL  PROBLEMS  IN  PERCENTAGE. 

(449,  page  261.) 

Ex.  5.  Ans.  70.65  Ib.  Ex.  6.     An*>  240  mi. 

Ex.  8.  Ans.  919  men.  Ex.  9.     756x1-25=945,  Ans. 

Ex.  12.  iX,fo=Tfo>  Ans- 

Ex.  13.  14f  %=4;  VX4 

Ex.  14.  $375x.05=$18.75, 

Ex.  15.  $536+$450+$784=$1770,  debts; 
$1770X.54=$955.80,  ^s. 

Ex.  16.  15  %+5  %+6  %+8  %=34  %=.34; 
$1500  X-  34—  $510,  4*w. 

Ex.  17.  $3500X3=$10500,  received  in  3  years; 

10  %+l2  %+lS  %=40%;  $3500X.40=$1400, 
spent  in  3  years;  $10500—  $1400  =$9  100,  Ans. 

Ex.18.  1.00—  .25=.75;  .75X.30  =  .225; 
.25+.225=.475  drawn; 
.475X-10=.0475  deposited  again; 

1.00—  .4275  =.5725  in  bank; 

$6000  X-  5725-4-13435,  Ans. 

Ex.  19.  2  %+3i  %+2  %+2£  %+l^  %+2|  %+4  % 

+3  %=21  %  entire  gain; 
-  4  %      «      loss; 


17  %  net  profits; 
$5400  X-  17  =$918,  Ans. 

(45O,  page  263.) 

Ex.  1.     $21.60-5-$720=.03=r:3  %, 
Ex.  2.  234-^-1560=.15=15 
Ex.  3.  49-f-980=.05=5  %,  ^TW. 

(261-263) 


126  PERCENTAGE. 

Ex.  4.  £320  10  s.=£320.5;  £25  12.8  s.=£25.64j 
25.64--320.5=.08=:8  %,  Ans. 

Ex.  5.     5  gal.  3  qt.  =  5.75  gal.; 

5.75-r-46=.125=123|  %,  Ans. 

Ex.  6.     5.495-5-7.85=.7  =  70  %,  Ans. 
Ex.  7.     f-r-T<t=f  =  .75=75  %,  Ans. 
Ex.  8.     ,&-H=TV*=.21=i21  %, 
Ex.  9.     80--560  =  4-  14f 
Ex.  10.  26.01--578:=.045=4i 
Ex.  11.  $145.836-v-$972.24=:15 
Ex.  12.  448-f-5600=z8  % 
Ex.  13.  4914-f-7560=:65 
Ex.  14.  455-v-2600=17^  %,  Ans. 
Ex.  15.  720  bar.—  288  bar.  =432  bar.  unsold; 
432-f-720=60 

Ex.  16.  |Xi%=A;  ^5 
Ex.  17.  $23,243,822.38 
14,712,610.21 


$37,956,432.59^-183,751,511.57 

—  .4532+  =45  1  %  nearly,  Ans. 

Ex.  18.  165x5=825,  aggregate  number  of  questions  asked; 
130+125+96+110+160=621,  number  answered; 
621-5-825=  .7527  =75.27  %,  Ans. 

(451,  page  264.) 

Ex.  1.  18-s-.25=72,  Ans. 

Ex.  2.  54-r-.15=360,  Ans. 

Ex.  3.  17.5-j-.02i  =  750,  Ans. 

Ex.  4.  2.28--.05=45.6,  J.ns. 

Ex.  5.  414-5-1.20=345,  Ans. 

Ex.  6.  6119-5-1.055=5800,  Ans. 

(263,  264) 


GENERAL   PROBLEMS.  127 

Ex.  7.     .43-r-.71f  =  .6,  Ans. 

Ex.  8.    $18.75-v-.025=$750,  Ans. 

Ex.  9.     31  j--r-.31|  =100,  Ans. 

Ex.  Id.  $4578-r-.84=$54ft),  ^ws. 

Ex.  11.  1.00—  .30=:.70  %,  sold;  3150-r-.70=4500,  Ans. 

Ex.  12.  .40X-131  —  .05-J  for  the  carriage; 
$116-3-.05i=$2175,  ^«8. 

Ex.  13.  $147.56-*-.13£  =$1106.70,  A's; 
$1106.70x.04f=$51.646; 
$51.646-*-.08  =1645.575,  B^s; 
$1106.70-  $645.575=$461.125,  Ans. 

Ex.  14.  100  %—  4  %—  96  %,  left  after  the  battle; 
5  =  .048=4f  %9  died  of  wounds; 
4  %=|  %=.008,  difference; 


Ex.  15.  A  has  |  and  B  -|  of  the  prize; 

1.00—  .40=.60;  |  XT6oV=-45  of  prize,  A'sremainder^ 
1.00—  .20=.80;jX^ft=.20       «      B's       « 
.45—  .20^.25;  $1950--.25=$7800,  Ans. 


5  page  265.) 

Ex.  1,     1.00+.15=1.15;  644-^1.15=560,  Ans. 
Ex.2.     1.00+.04=1.04;  $815.36^-1.04=$784,  Ans. 
Ex.  3.     1.00+12=1.12;  $3800-f-1.12=$3392.86—,  Ans. 
Ex.  4.     1.00+.10=1.10;  39600-5-1.10=36000,  -4ns. 
Ex.  5.     1.00+1.08=2.08=208  %  of  last  year's  crop,  raised 

in  the  two  years;  hence, 

5200  bu.--2.08=2500  bn.,  Ans. 
Ex.6.     1.00+1.05=2.05; 

$6970^-2.05=$3400,  1st  year's  crop; 

«3400X1.05=$3570,  2d      «        " 
(264-266) 


128  PERCENTAGE. 

Ex.  7.  If  the  number  be  increased  8  %,  the  amount  will  be 
108  %  of  the  number;  and  if  this  amount  be  in- 
creased 7  %,  the  whole  amount  will  be  107  %  of 
108  %,  or  1.08X1.07=2=1.1556  times  the  number; 
hence  86.67-5-1.1556=75,  Ans. 

Ex.  8.     Since  he  gained  1^  %  in  measure  and  5  %  in  price, 
he  sold  1.015  times  the  number  of  bushels  bought, 
at  1.05  times  the  buying  price;  hence   he  received 
1.015x1.05=1.06575  times  the  cost; 
and$4910.976-f-1.06575=$4608,  Ans. 

Ex.  9.     1.00=A's  proportion; 
1.06=B's 
1.04=C's         " 


3.10r±all  their  money ; 
$11160-r-3.10:=$3600,  Ans. 

Ex.  10.  .The  material  cost  200  %  of  the  labor;  the  cost  of 
labor  increased  9  %,  would  be  109  °/0  of  the  labor; 
the  cost  of  material  increased  6  %;  would  be  212  % 
of  the  labor ;  hence  the  cost  of  the  house,  thus  in- 
creased, would  be  109  %+212%=321  %  of  the 
cost  of  labor;  and  $1284-v3.21=$400,  cost  of  labor; 
$400X2  =$800  «  material: 


$1200,  Am. 

(458,  page  267.) 

Ex.  1.     1.00— .10=.90j  504--.90=560;  Ans. 
Ex.  2.     1.00— .08=.92;  $4.37~.92=$4.75;  Ans. 

Ex.  3.     1.00— .15=.85;  40  bu.  31  pk.=40.8  bu.; 
40.8  bu.-^.85=48  bu.,  Ans. 

Ex.  4.     1.00— .36=.64;  224  A.-r-.64=350  A.,  Ans. 
(266,  267) 


GENERAL   PROBLEMS.  129 

Ex.  5.     1.00— .65=.35 ;  $2590--.35=$7400;  Am. 
Ex.  6.     Having  drawn  out  20  %  of  his  deposit,  he  had 

100  % — 20  %=80  %  of  his  deposit,  in  bank;  and 

80  %  of  his  deposit  is  80  %  of  80  %  of  his  fortune; 

.80X.80=.64;  $5760-=-.64=$9000,  An*. 

Ex.  7.     100  %— 12  %=88  %  of  his  share  sold  to  B;  and 
f  XT8D8o=.77  of  the  ship  sold  to  B; 

$20020-r-.77=$26000,  Ans. 

Ex.  8.     1.00— .10 =.90,  left  after  the  first  battle ; 

.90X-10—09;  .90— .09  =  .81,  left  after  2d  battle; 
6480-r-.81=8000  men,  Ans. 

Ex.  9.  B's  asking  price  was  1.00+.50=1.50=150  %  of  A's; 
1.00— .20 =.80 ;  that  is,  A's  reduced  price  was  80  % 
of  his  asking  price;  1.50X-30=.45;  1.50— .45 
=1.05;  that  is,  B's  reduced  price  was  105  %  of 
A's  asking  price;  hence  the  sum  of  the  reduced 
prices  was  80  %-f-105  %=185  %  of  A's  asking 
price;  and  $148-^-1.85=880,  A's  asking  price ; 

880X1.50=8120,  B's     "        " 

NOTE. — In  this  solution  A's  asking  price  is  the  lase  to  which  all 
the  rates  are  referred. 

Ex.  10.  Of  the  whole  sum  expended,  |  was  for  wheat,  ^  foi 
corn,  and  ±  for  oats :  hence  we  have 

'  O  f 

6  %  of  1=2  %,  cleared  on  wheat; 
3  %  of -1=1  %,       "        "  corn;. 
17  %  of  i=5|  %,  lost       «  oats; 

2|  %,  net  loss; 
1.00— .02f  =  .97|,  saved; 
$2336-7-.97|=$2400,  whole  sum  expended  ; 
$2400-^-3 =$800,  sum  expended  for  each  kind. 

(267) 


130  PERCENTAGE. 

APPLICATIONS   OF   PERCENTAGE. 
COMMISSION. 

(462,  page  270.) 

Ex.  4.     88600  X- 02  J=8193.50,  Ans. 
Ex.  5.     $750.75X.03f=828.15+,  Ans. 
Ex.  6.    81.40x3500=84900 
$  .74X3600=2664 

$7564X-02i=$170.19,  Ans. 

Ex.  7.     $.23^X2000=$  470 
.22  X5650=  1243 
.23  X  450=     103.50 
.21  X  650=     136.50 


$1953.     X.08|=$162.75,  Ans. 
Ex.8.     $25372X.06;1=$1585.75,  commission; 

132.     ,  storage; 

$1717.75,  entire  charges; 
$25372— $1717.75=823654.25,  Ans. 

Ex.  9.     $25x40=11000,  cost  of  apple  trees; 
50X20=  1000,    "     "   pear      " 
20x16=     320,    "     "   peach    « 
50X18=    900,    «     •<   cherry  " 
50 X  5=     250,    «     «   plum     « 

$3470,  amount  of  sale; 
83470  X  .30=11041,        commission ; 
$  203.50,  expended; 


$3470— $1244.50=12225.50,  proceeds,  Ans. 

Ex.  10.  $785X-82=$643.70,  sum  collected; 

3.70 X-05r=$32. 185,  commission,  Ans. 

(270) 


COMMISSION.  131 

Ex.  11.  81.25X4000=85000,  cost; 

81.50X4000=66000,  receipts; 
$6000X-03=$180,  commission; 
$180+$415=$595 ;  86000—8595=85405,  net 
proceeds;  $5405— $5000 =$405,  Ans. 

Ex.  12.  $63-4-$1260=.05=5  %,  Ans. 

Ex.  13.  $4.50X264=$1188,  base  of  commission ; 

«74.25--$118S  =  .0625=6j  %,  Ans. 

Ex.  14.  $7850— $7732.25=1117.75,  commission; 

$117.75-^-$7850=.015=li  %,  Ans. 

Ex.  15.  $.121X2SOOO=$3500,  receipts; 

$35.36 +$10.50= $45.86,  charges; 
$3500—  83252.89=1247.11,  com.  and  charges; 
$247.11— $45.86 =$201.25,  commission  ; 
8201.25-=-83500=.0575=5j  %,  Ans. 

Ex.16.  $5635+$115=$5750,  receipts; 

$115-i-$5750  =  .02=2  %,  Ans. 

Ex.  17.  $22.40-v-.04=$560,  Ans. 
Ex.  18.  $6.80-^-.08=$85,  Ans. 

Ex.  19.  8255-r-1.02=8250  to  be  invested; 
$250=$.15=1666§  yd.,  Ans. 

Ex.  20.  $7.50x860 =$6450,  receipts; 

$6450 X.02£= $161.25,  commission  on  sale; 
$6450— 8161.25=86288.75,  net  proceeds  of  sale; 
$62S8.75-v-1.015=$6195.81+,  Ans. 

Ex.  21.  $.06X24000 =$1440,  receipts  for  pork; 

$1440 X -05 =$72,  commission  on  sale  of  pork; 
$1440— $72  =  $1368,  net  proceeds; 
$3000+$1368=$4368; 

B-f-1.05=$4160  to  be  invested; 

(270,  271) 


132  PERCENTAGE. 

$-1368 — $4160 =$208,  commission  for  investing; 

$72+$208 =$280,  Ans. 

Ex.22.  1.00— .06=.94;  $3290--.94=$3500,  Ans. 

Ex.  23.  $6290+$500=$6790,  receipts,  less  the  commission ; 
1.00— .03  =  .97;  $6790--.97  =  $7000,  receipts; 
$7000-r-500=$14,  Ans. 

Ex.  24.  $3500X-04=$140,  purchase  commission; 

$3500-f  $140  =  $3640,  net  proceeds  of  auction  sale; 
1.00— .09  =  .91;  $3640-h,91=$4000,  Ans. 

Ex.  25.  Since  the  purchase  commission  was  2  %  of  the  price 
of  the  city  lots,  the  net  proceeds  of  the  sale  of  cotton 
were  102  %  of  the  cost  of  the  city  lots ;  and  as  the 
proceeds  divided  by  1  minus  the  rate  equal  the  sale, 
we  have  102  %-=-.97  =  105£f  %;  that  is,  the  sale 
of  the  cotton  was  105^|  %  of  the  price  of  the  city 
lots;  now,  as  the  difference  between  the  receipts 
from  the  cotton  and  the  price  of  the  lots  is  the  whole 
commission,  $265  is  1.05Jf — 1.00=5^  %  of  the 
price  of  the  lots ;  therefore, 

$265~.05f!  =  $5141,  Ans. 

Ex.26.  1.— .0075^.9925; 

$5000-4-,9925=$50377.83+,  Ans. 


STOCKS. 
STOCK-JOBBING. 

(48O,  page  275.) 
Ex.  4.     $1500X1-12  =  $1680,  Ans. 

Ex.  5.     81.35-4.0175= $1.3325,  proceeds  of  $1; 
$2000X1.3325— $2665,  Ans. 
(271-275) 


STOCKS.  133 

Ex.  6.     $.'8075+$.005=$.8125,  cost  of  $1  of  stock; 
$12000  X.  8125=19750,  Am. 

Ex.  7.     1.075+.0025=1.0775; 

$3600X1.0775=13879,  An*. 

Ex.  8.     1.00—  .60f=.39£=.39125; 

$21910--.39125=$56000=560  shares,  Ans. 

Ex.  9.     $40150-  $40000=$150,  premium; 


Ex.  10.  $.94875  +  $.0125=1.96125,  cost  of  $1; 

$48447-*-.96125=$50400=504  shares,  Ans. 

Ex.  11.  $6  X830=$4980,  receipts;  $4980  X.05=$249,  com- 
mission; $4980—  $249  =  $4731,  net  proceeds  of  sale; 
.82f-f.00|=.83; 
$4731~.83=$5700=57  shares,  Ans. 

Ex.  12.  $500X18=49000,  par  value  of  stock; 
.02+.28=.30,  rate  of  loss; 
$9000X.30=$2700,  Ans. 

Ex.  13.  $3600  X   .95=$3420,  value  of  railroad  bonds; 
$2700X1.03=  2781,  value  of  bank  stock; 


$  639,  Ans. 
Ex.  14.  ^f  *|£4.z=520  share's,  Ans. 

Ex.  15.  $1200XL05=$1260,  purchase  price; 
$1260— $96=$1164,  selling  price; 
$1164-=-$1200=.97=97  %,  price  per  share,  Ans. 

Ex.  16.  $64000xl.02=$65280,  cost; 

$65280+$2560=$67840,  to  be  sold  for; 
$67840-*-$64000=1.06=106  •%,  Ans. 
(275,  276) 


134  PERCENTAGE. 

Ex.  17.  $25000xL03=$25750,  cost  of  mining  stock; 
$15000  X   .95=  14250,    "     "  railroad     " 


$40000,  money  invested ; 

$40000--.80=:$50000,  par  val.  of  N.  Y.  C.  stock; 
$50000 X-85=$42500,  money  received; 
$42500— $400.00 = $2500,  An*. 

Ex.  18,  $750-^.03 =$25000 =250  shares,  An*. 

Ex.  19.  105  %— 103  %=2  %,  gained; 
$240-^.02  =  $12000,  Ans. 

Ex.  20.  4  %+5£  %=9£  %,  loss; 

$760-;-.095=$8000Z:80  shares,  Ans. 

Ex.21.  1.06+.0125=1.0725; 

$6864-r-1.0725=$6400,  par  value  of  bonds  bought; 
1.12— .015=1.105; 
$6400x1.105=17072,  stock  sold  for; 

$7072— $6864=1208,  Ans. 

Ex.  22.  7  %—  3  %=4  %,  gain; 

$480-r-  .04=$12000,  par  value  of  stock; 
$12000xL03=$12360,  money  invested,  Ans. 

Ex.23.  .04+.005=.045;  1.00— .045=.955; 
$4775-f-.955=$5000  =  50  shares,  Ans. 


INSTALLMENTS,  ASSESSMENTS  AND  DIVIDENDS. 

(487,  page  277.) 
Ex.  3.     $5600X-08=$448,  Ans. 
Ex.  4.     $15000X-06=$900=     9  shares, 
$15000=150      « 

159  shares,  Ans. 
(276,  277) 


STOCKS.  135 

Ex.  5.     $6500  X  .15  =  $975,  Ans. 
Ex.  6.     $600-r-.04=$15000,  Ans. 

Ex.  7.     $99280— $56400 =$42880,  dividend  : 
$42880--$536000=.08  =  8  %,  Ans. 

Ex.  8.     $56000 X-06=$3360,  dividend; 
$3616— $3360 =$256,  Ans. 

Ex.  9.     $450000X-05f =$25000,  interest; 

$25000+$217621= $242621,  disbursements  •; 
$407399— $242621=$164778,  net  earnings; 

$164778— $78 =$164700,  dividend; 
$164700--$1830000=.09,  rate  of  dividend; 
$3000X-09=$270,  Ans. 

Ex.  10.  $540000 
400000 

$940000,  entire  cost  of  the  road ; 
800000,  capital  stock  available ; 

$140000,  to  be  assessed ; 
$140000-v-$800000=.17^=17J  %,  Ans. 
Ex.  11.  $156753.19-r-.0525= $2,985,775  nearly,  Ans. 

Ex.  12.          $574375.25 
643672.36 


$1218047.61,  gross  earnings; 
651113.53 


.08)  $566934.08,  net  earnings; 

$7086676,  capital  stock,  Ans. 
Ex.  13.  504-5-1.05=480,  Ans. 

Ex.14.  .06-h  J5=.08;  hence,  a  6  %  dividend  invested  at  75 
%  is  equal  to  an  8  %  dividend  invested  at  par; 
(277,  278) 


136  PERCENTAGE. 

Therefore, 

$16200-r-1.08=$15000,  his  stock  before  the  div.; 
'   $16200— $15000=11200,  increase  of  his  stock; 
$1200  X -75 =$900,  value  of  the  increase  at  75  %, 
or  the  money  received  in  the  dividend,  Ans. 

Ex.  15.  .05 +.05 =.10,  annual  rate  of  dividend; 
$28000X-10=$2800,  dividend; 
$2800+$2950=$5750,  Ans. 


STOCK   INVESTMENTS. 

(490,  page  279.) 

Ex.  1.     $35374.80-^-1.025  =  $34512,  stock  purchased  ; 
$34512X.05=$1725.60,  income; 
$1725.60—  $1000=$725.60,  Ans. 

Ex.2.     .95i+.00|  =  .96;  1.12+.00£=1.125; 

$48000-f-2=  $24000,  invested  in  each  kind  of  stock; 
$24000-r-.96=$25000,  purchased  of  5  %  stock; 
$24000-f-1.125=$21333.33|,  "  6  %  " 

$25000  X  .05        =  $1250,  income  from  the  5's  ; 
$21333.33|  X-06=$1280,     «          «      «   6?s; 

$2530,  Ans. 

Ex.  3.     $90000  X-  88  =$79200,  market  value  of  bonds; 

$79200-7-.625=$126720,  capital  stock  purchased  ; 
.03+.03^  =  .06^,  annual  rate  from  capital  stock; 
$126720  X  -06^  =  $8236.80,  income  of  capital  stock  ; 
$90000  X  -7  =  6300  «        «  bonds; 


$1936.80,  Ans. 
(278-280) 


STOCKS.  137 

Ex.4.     $32300-f-.85=$38000,  purchase  of  6;s; 
$32300-v-.95=$34000,        "          "  7's; 
$38000 X-06=;$2280,  income  from  6's; 
$34000X.07=$2380,      «        "     7's; 


$100,  Ans. 

(491,  page  280.) 
Ex.  1.     $1200-^-.08= $15000,  stock  required; 

$15000  X-855=$128|5,  investment,  Ans. 

Ex.  2.     $960-f-.06=$16000,  stock  required; 

$16000  X- 84 = $13440,  investment,  Ans. 

Ex.  3.     $630-r-.06=$10500,  par  value  of  the  Virginia  6's; 
$630-^.05= $12600,  U.  S.  5  per  cents  required; 
$12600X1. 03 =$12978,  cost  of  U.  S.  5's; 
$10500  X   -85=     8925,  price  of  Virginia  6's; 

$4053,  Ans. 

Ex.  4.     $840~.07=$12000,  California  7's  requited; 
$12000 X.96=$11520,  cost  of  Cal.  7's; 
$11520-r-.60=$19200=192  shares,  Ans. 

Ex.  5.     Suppose  $1.00  to  be  invested  in  each  kind  of  stock; 
then  we  should  have 

$1.00-r-1.04:=$igJ,  U.  S.  5  per  cents  purchased; 
$1.00-^.95=$  W0,  Md.       «       «  « 

o^  entire  stock  purchased; 
,  income  by  investing  $1  in  each 
kind  of  stock;  hence  $2487.50-^^= $24700,  Ans. 

(492,  page  281.) 

Ex.  1.     .06-s-1.08  =  .05f =5f  %,  Ans. 
Ex.  2.     .06-*-.84=. 074=74  %,  Ans. 
(280,  281) 


138  PEKCENTAGE. 

Ex.  3.     1.34i+.OU=1.36; 

.085--1.36=.0625=6|  %,  Ans. 

Ex.  4.     .05-7-.  70—  .074?  rate  on  investment  in  5's; 

.OG-r-.80=.07£,    "  "  "  6's,  better. 

Ex.  5.     .08-f-.12=.06|,  ) 

05-1-75  _  06^    f      es  on  mvestmen^s  e(pal. 

3  )    J 

Ex.  6.     $1722996--$24182400=.07|,  rate  of  dividend; 


(493,  page  282.) 
Ex.  1.     .05^-.08=C2^  %,  Ans. 
Ex.  2.     .06-^.0625=96  %,  purchase  price; 
100%—  96  %=4  %,  Ans. 

Ex.  3.     .07-f-.06=1.16|,  market  price; 
1.16|—  1=.16|=16|  %,Ans. 

Ex.  4.     .06-^.08^  =  72  %,  An*. 


PROFIT  AND  LOSS. 

(495,  page  284.) 
Ex.  4.     $7650X.20=$15307  advance; 

$1530— $480=$1050,  Ans. 

Ex.  5.     $.15X320=$48;  cost;  $48x.025=J1.20,  Am. 
Ex.  6.     $3.50X30=1105,  purchase  price; 

$105  X  .90=894.50,  selling      « 

$94,50 X-05=44.72£,  commission; 

$105+$4.72|4-«5.38=$115.10^,  whole  outlay; 

$115.101— $94.50=$20.60|,  Ans. 

Ex.  7.     l+.33|=1.33i=l| ;  $.50xli=$.66|,  Ans. 
(281-284) 


PROFIT   AND   LOSS.  139 

Ex.  8.     $4.93— $4.25=$.68,  gain  per  quintal; 
$.68--$4.25=.16=16  %,  Ans. 

Ex.  9.     $59 X- 20 =$11.80,  advance  of  20  %  ', 

859+811.80+84.72=875.52,  to  be  sold  for; 

$75.52-^944=$.08,  Am. 

Ex.  10.  $157.50X.05=$7.875,  whole  loss; 

$157.50— $7.875=1149.625,  sold  for; 
$149.625-r-$3.325=45  gallons  sold; 
63  gal.— 45  gal.  =  18  gal.,  Ans. 

Ex.  11.  106  A.  3  E.  30  P.  =  106.9375  A.; 

$96X106.9375=$10266,  selling  price; 
$10266-7-1.18=88700,  Ans. 

Ex.  12.,  821.12X36.840  =  8778.0608,  sold  for; 
8778.0608-=-1.28  =  8607.86,  cost; 
$17X36.840=$626.28,  Value  at  $17  per  M., 

$626.28— $607.86=$18.42  gain,  Ans. 

Ex.  13.  1— .04=. 96=96  %,  purchase  price; 
1— .28  =  .72  =  72  %,  selling         « 
96  %—  72  %=24  %,  difference  of  prices; 
.24-f-.96=.25=25  %,  Ans. 

Ex.  14.  $7050--.94=$7500,  cost; 

87500X1.12^=88437.50,  Ans. 
Ex.  15.  $182-^.875 =$208,  cost  of  second; 

8208-5-1.30=8160,    "     «  first. 

Ex.  16.  15  .  %  on  |  is  I  of  15     %  =  3|  %  on  the  whole; 

18 j  %  on  i  is  j  of  18|  %=6|  %  "     «       " 

20     %  on  J  is  J  of  20     %=3|  %  «     «        « 

83f  %  on  i  is  i  of  33|-  %=8|  %  «     «       « 

21|  %;  rate  of  grain  j ; 
86840 X-21|= $1482,  entire  gain,  Ans. 

(284,  285) 


140  PEKCENTAGE. 

Ex.  17.  100  %  cost; 

1.00  Xl-50— 150    %,  after  1  year ; 
1.50  XL50=225    %,     "    2  years; 
2.25  X  1.50=337^  %,     "    3     " 
3.375xl.50=506|  %,     "    4    « 
$12000-r-5.0625=$2370.3751T, 


Ex.18.  $1.80  X  600=$1080j 
$1.62^xl200=$1950}  = 

$1.25  X  200=     250 


$3280,  cost ; 

$3030x1-20=83636,  receipts  from  first  two  kinds; 
$1.20  X200  =$  240,      "          "     third          " 


$3876,  total  receipts; 
$3876X-05=$193.80,  commission; 
$3876—  ($193.80+$254.60)  =$3427.60,  proceeds; 
$3427.60—  $3280  =$147.60,  gain; 
4|  %,  Ans. 


Ex.  19.  100  %—  20  %  =  80  %,  buying  price; 
100  %—  16  %=84  %,  selling    « 

4  %-i-SQ  %=5  %, 

Ex.  20.  Suppose  he  sells  1  share  of  $100;  then  $100X-05 
=$5.00,  annual  income  parted  with; 
$78-7-1.04=$75,  stock  purchased  ; 
$75  X  -06  =$4.50;  annual  income  in  return; 
$5.00—  $4.50=$.50     «          «      lost; 
$.50-5-85.00=.10=10  %,  Ans. 

Ex,  21.  $125X12=$1500,  received  for  each  half; 
•1500-4-1.25  =$1200,  cost  of  one  half; 
$1500-f-  .75=$2000,  «    "  the  other  half; 

(285) 


PROFIT  AND  "LOSS.  141 

$1200+$2000=$3200,  cost  of  the  whole; 
$1500X2         =  3000,  received  for  the  whole ; 

$  200  lost,  Ans. 

Ex.  22.  $30xl.20=$36,  selling  price; 

$36-77-. 75— $48,  asking  price,  Ans. 

Ex.  23.  He  asked  120  %  of  the  cost,  and  sold  at  1.00 -.20 
=80  %  of  the  asking  price.  Hence  120X-80 
=.96;  1.00— .96=.04=4  %  loss,  Ans. 

Ex.  24.  $.05  X216X200=$2160,  cost  of  N.  O.  sugar; 
$.05JX200X560=  6440,    "    «  W.  I.     « 


0,  entire  cost ; 
$2160X-99=  $2138.40,  rec'd  for  N.  0.  sugar; 

$6440xl.00||=       6496     ,      "    «  W.I.     " 

$8634.40^      "    "  the  whole; 
$8634.40— $8600=$34.40,  gain; 
$34.40-s-88600=.004=f  %  gain,  Ans. 

Ex.  25.  $31.50-f-.70=$45,  prime  cost  of  the  whole; 

$45-f-63=$.71f ,  purchase  price  per  gallon,  Ans. 

Ex.  26.  1.00— .07|=.92|=. 924,  selling  price;; 

,924-f-1.05=.88,  purchase  price; 

1.00— .88=.  12=12  %,  Ans. 
Ex.  27.  $.025~.08=$.31|,  Ans. 
Ex.  28.  $.75-rU875=$4.00,  cost; 

$4.00 X. 31^=81.25,  advance  on  cost-; 

$1.25—8.75=8.50,  Ans. 

Ex  29.  30  %  of  |=.18  of  the  whole; 
5  o/0  of  |=.02      «  *      « 

$720-f-.16=$4500,  investment; 
(285,  286) 


142  PERCENTAGE. 

30  %  of  f-5-.12,  of  the  whole; 
5  %  of  | =.03,      "        " 

$4500X-09=$405,  Ans. 

Ex.  30.  His  asking  price  was  136  %  of  the  cost,  and  his 
selling  price  was  100  %— 16  %=84-  %  of  his  ask- 
ing price,  or  1.36  X  .84=1.1424=114-^  %  of  the 
cost.     Hence  the  gain  was  14^  %  of  the  cost;  and 
$740.48-r-.1424=:$5200,  the  cost; 
85200x1-36=87072,  the  asking  price; 
$5200X1.1424=15940.48,  sold  for,  Ans. 

Ex.  31.  Since  his  gain  on  |  was  |,  we  have  f-i-f =.60=60  %, 
the  rate  of  gain,  Ans. 

Ex.  32.  ($.375X84 X,4)-f$7.50 =$133.50, whole  cost; 
$133.50XL25  =  $166.875,  net  avails  required; 
$166.875-H»99  =  $168.56,  to  cover  the  collector's  fee  ; 
$168.56-7-.96=$175.58,  whole  sum  to  be  obtained 
to  cover  bad  debts  and  collector's  fee ;  but  84  gal. 
X4=336  gal.  bought;    336  gal. X-05= 16.8  gal, 
leakage  and  waste;  336  gal.— 16.8  gal.=319.2  gal. 
left  for  sale.     Hence,  $175.58-4-319.2=155+,^. 


INSURANCE. 
FIRE   AND.  MARINE   INSURANCE. 

(5O3,  page  288.) 

Ex.  3.    $5860X-015=$87.90,  Ans. 
Ex.  4.     $860  X  .005  =  $4.30,  Ans. 
Ex.  5.  83500X.01|=$43.75,  Ans. 

(286-  -288) 


INSUEANCE.  143 

Ex.  6.  $10000X-0225=$225,  premium; 
$10000— $225=$9775,  Am. 

Ex.  7.     $6000X.01^=$75,  premium; 

$6000— <$75+$5.50)=$5919.50  covered; 
$10000— $5919.50=14080.50,  An*. 

Ex.  8.     $12000xf=$9600,  policy; 

$9600X-00|=$72,  premium; 
$9600— $72 =$9528,  covered  by  insurance; 
$12000— $2000 =$10000,  whole  loss; 
$10000— $9528  =  $472,  owner's  loss,  Ans. 

Ex.  9.     $36000X-02i=$900,  premium  received; 
$1 8000 X- 03  =  540,         "       paid  out; 


$360,  An*.' 
Ex.  10.  $12-j-$800=.015=l^  %,  Ans. 

Ex.  11.  $107.25-r-.0325=$3300,  policy; 

$3300-f-.80=$4125,  cost  of  wheat; 
$4125-;-500=$8.25,  price  per  barrel,  An*. 

Ex.  12.  $280-v-$16000=.0175=l|  %,  Ans. 

Ex.  13.  $20000  X-OOf  =  $150 
0=  150 


$50000  $300-f-$50000=.006=|  %,  Ans, 

Ex.  14.  $46.75--.01375=$3400,  policy; 
$3400X2=$6800,  An*. 

Ex.15.  1—  .01125=.98875;  $7910-f-.98875=$8000,  An*. 

Ex.16.  1—  .02f  =  .974;   $27320-f-.97|=  $28000,  policy; 

$28000—  $27320=$680,  Ans. 

Ex.  17.  $122.50-^.04|=$1400,  policy; 
$1400-*-f  =$4480,  An*.  * 
(288,  289) 


144  .  PERCENTAGE. 

Ex.  18.  On  each  $1  which  the  house  is  worth,  he  obtains  an 
insurance  of  $.75,  for  which  he  pays  during  the  5 
years,  $.75X.015X5=$.05625;  hence  the  insur- 
ance will  cover  $.75 — $.05625=$.69375;  and  his 
loss  on  the  $1  will  be  $1— $.69375=1.30625;  there- 
fore, $2940-^.30725 =$9600,  Ans. 

Ex.  19.  1.00-^.025=40  years,  Ans. 

Ex.  20.  .025x1—015,  the  per  cent,  of  the  policy  paid  for 
re-insuring;  hence  .0225 — .015 =.0075; 
and  $72~.0075=$9600,  Am. 

Ex.  21.  The  two  companies  together  must  lose  1 — .042 & 
=.9575  of  the  policy;  and  since  the  Manhattan 
Company  becomes  responsible  for  .50  of  the  policy, 
and  receives  . 50  X -03  =  .015  of  the  policy  in  pre- 
jaaium,  it  must  lose  .50— .015=.485  of  the  policy; 
therefore,  the  Commercial  Insurance  Company  will 
lose  .9575 — .485 =.4725  of  the  policy;  and  conse- 
quently .485— .4725=.0125  of  the  policy  is  $1350; 
hence  $1350-4-.0125= $108000,  the  policy; 
$108000-f-  |  =$144000,  value  ship  and  cargo; 
$108000X.9£75=  103410,  covered  by  insurance; 

$40590,  Ans. 


LIFE  INSURANCE. 

(5O9,  page  293.) 

Ex.  2.     50 — 30+1=21,  number  of  payments; 

$2000X-029928=$59.856,  annual  payment; 
$59.856X21=$1256.976,  whole  sum  paid; 
$2000— $1256.676=$743.024,  Ans. 

Ex.  3.     $3000X-046417=$139.25+,  Ans. 
(289-293) 


INSURANCE.  145 

Ex.  4.     52 — 36+1=:  17,  number  of  payments; 

§1500 X- 031971 =$47.9565,  annual  payment; 
$47.9565Xl7=$815.26,  sum  of  'payments ; 
$1500— $815.26=4684.74,  Am. 

Ex.  5.     72 — 544-1=19,  number  payments  made ; 

65—54+1=12,       «  "       ceasing  at  65; 

$3500X.055067X19=$3661.95+,  money  paid; 
$3500X-078017X12=$3276.71+ 


5.24,  Ans. 

Ex.  6.     40—30+1=11  No.  pa/ts  on  policy  issued  at  30; 
40—24+1=17  "        "      "      "          "      "  24    ; 
$1200X.09526X11=$1257.432,  policy  issuedat30; 
$1200X- 05544x17=11130.975,      «        «    "24; 


$126.456,  Ans. 

Ex.  7.     49 — 37+1=13,  number  of  payments; 

.07325— .04525 =.028,  difference  in  annual  rate  %  ; 
$750 X- 028X13  =  $273,  Ans. 

Ex.  8.  $100-5-81.7296=58—,  A's  number  of  payments, 
requiring57  years;  hence  20+57=77  years,  A'sage; 
$100n-$2.3023=44— ,  B's  number  of  payments  re- 
quiring 43  years;  hence  30+43=73  years,  B's  age. 

Ex.  9.     60 — 45+1=16,  number  of  payments ; 
$1000X-06686X16=$1069.76,  Ans. 

Ex.  10.  $600X-037142=$22.285+,  1  payment; 

$600— $421.72=$178.28,  whole  premium  paid; 

$178.28-7-422.285=8,  Ans. 

Ex.  11.  40—29+1=12,  number  of  payments;  $.022346 

X 12 =$.268152,  whole  premium  on  a  policy  of  $1 : 
$1— $.268152=$.731848,  secured  on  each  $1 ; 

$1829.62--.731848=:$2500,  Ans. 

(293,  294) 
13 


$35.62,  ^4ns. 

Tax  on  < 

tt     a 

5900        is 
80        « 

$3.60 
.32 

u     « 

7        ** 

.028 

U         (( 

1  poll  * 

.50 

146  PERCENTAGE. 

TAXES. 

.     (514,  page  296.) 

Ex.  2.     Tax  on  $8000  is  $32.00 

«      "       500  "  2.00 

"     "         30  "  .12 

«     "           3  polls  "  1.50 


Ex.3. 


$4.448,  Ans. 

Ex.  4.     $.25X2156=r$539,  amount  of  poll  tax, 

$6319— ($654+$539)=$5126,  property  tax; 
$5126-r-$1864000=  .00275,  rate  of  taxation  for  town 
purposes;  .00275+.0015+.0005=.00475,  full  rate; 
$32560 X. 00475= $154.66,  A's  property  tax; 
$.25X3=         .75,   «    poll  tax; 

$155.41,  Ans. 

J£x.  5.     $16840  X- 00475=$79.99,  Ans. 
Ex.  6.     $5561.50-^.98  =  15675,  Ans. 

Ex.  7.     $9120-5-.95=$9600,  tax  to  be  raised; 

$9600-f-$1536000=.00625=f  %,  Ans. 
Ex.  8.     $1200+$57.65+$38.25=$1295.90,  whole  expense- 

$1295.90— $257.75=$1038.15,  tax  by  rate-bill; 

$1038.15-^9568=$.108502+,  rate  per  day; 

$.108502X46X4=$19.96+,  Ans. 

Ex.  9.     $1260.52-^.965=$1306.24+,  tax  to  be  assessed; 
$1306.24-f-.00325=$401920,  base  of  taxation,  or 
valuation  of  property,  Ans. 
(296) 


GENERAL  AVERAGE.  147 

GENERAL  AVERAGE. 

(51 85  page  298.) 

Ex.  1.  Losses.  Contributory  Interests. 

Jettson,  $6300  Vessel,  $25000 

Repairs,  less  |,        1000  Freight,  less  |,       2800 

Cost  of  detention,      420  Cargo,  25200 

Total,  $7720  Total,  $53000 

$7720-r-$53000=.1456603-{-,  rate  of  loss; 
$25200 X-1456603 =$3670.64,  payable  by  Hayden  &  Co.; 
$6300      ,  receivable         "  " 


$2629.36,  balance  due      "  " 

Or, 

$27800 X- 1456603 =$4049.36,  payable  by  George  Law; 
1420      ,  receivable         "         " 


$2629.36,  balance  due  Hayden  &  Co. 

Ex.  2.  Losses.  Contributory  Interests. 

Jettson,  $1570  Vessel,                $28000 

Repairs,  less  -|,  180  Freight,  less  |,        1000 

Detention,  120  Cargo,                      5000 

Total,          $1870  Total,          $34000 

$1870-f-$34000=.055,  rate  of  loss; 
$29000 X.055=$1595,  contributed  by  ship  owners; 
$180+$120    =     300,  due  "         « 


$1295,  payable  by 

$750X.055=$41.25          «       «  C; 
$2400 X- 055 =$132,  contributed  by  A; 
$1400— $132 =$1268,  receivable  «    « 
$1850X.055=$101.75,  contributed  by  B; 
1.75;=$68.25,  receivable   "    « 
(298) 


148  PEKCENTAGE.    ' 

CUSTOM  HOUSE   BUSINESS. 

(529,  page  301.) 
Ex.  4.     $2780X.04=$111.20,  Am. 
Ex.  5.     $.055Xl200=$66,  cost;  $66X-15=$9.90,  Ans. 

Ex.  6,     112  Ib.  X  54=6048  Ib.,  gross  weight,  long  ton  measure; 
6048  lb.X-035=211.68  Ib.,  tare; 
6048  Ib.— 211.68  lb.=5836.32  Ib.,  net  weight; 
$.0875X5836.32  =  $510.678,  net  value; 
$510.678 X-15=$76.60-f ,  Ans. 

Ex.  7.     36  gal. X 50  =  1800  gal.,  gross  measure; 
1800  gal. X- 015 =27  gal.,  leakage; 
1800  gal. — 27  gal.  =  1773  gal.,  net  measure; 
$2.50X1773=$4432.50,  net  value; 
$4432.50 X. 30  =  $1329.75,  duty; 
$2.50X1800    =  4500        invoice; 
$1.30X50        =       65        freight; 
8.50,  cartage ; 

$5903.25,  Ans. 

Ex.  8.     120  doz.X- 02=2.4  doz.,  breakage; 

120  doz. — 2.4  doz.  =  117.6  doz.,  net  measure; 
£1.25XH7.6=$147,  net  value; 
$147X-24=$35.28,  Ans. 

Ex.  9.     $619.4/3--.19=$3260,  Ans. 

Ex.  10.  ,05|X  14X600 =$441,  invoice,  base  of  duty; 

$35.28-j-$441=.08  =  8  %,  Ans. 

Ex.  11.  $.30X  63X200=$3780 
$.05X500X150=  3750 


$7530X.24=$1807.20 
(301,  302) 


SIMPLE   INTEREST.  149 

$3.00X75=4225 


$425  X- 08=  34 

$1841.20,  Am. 

Ex.  12.  36  gal.X56=2016  gal.; 

$907.20-i-.30=$3024,  cost; 

83024-5-2016=81.50,  Ans. 
Ex.  13.  $132-j-.24=$550,  on  which  duty  was  reckoned; 

$550--.88=$625,  Ans. 

Ex.  14.  $1980.50— $12.24=$1968.26,  invoice  and  duty; 
$1968.26— $1654=$314.26,  duty; 
$314.26-r-$1654=.  19=19  %,  Ans. 

Ex.  15.  $823.20-j-.24=$3430,  net  value; 

$3430--$.25=13720=1143|  doz.,  Ans. 


SIMPLE   INTEREST. 

« 

(54O,  page  307.) 

Ex.  3.    Ans.  $.921.  Ex.  4.     Ans.  $5.885+. 

Ex.  6.     Ans.  $178.94—.  Ex.  8.     Ans.  $770. 

Ex.  12.  Ans.  $2.80.  Ex.  14.  Ans.  $195.765+. 

Ex.  15.  Ans.  $12.06+.  Ex.  16.  Ans.  $46.875. 

Ex.  18.  Ans.  $160.625.  Ex.  20.  Ans.  $28.34+. 

Ex.  23.  Ans.  $24.18+.  Ex.  24.  Ans.  $72.96. 

Ex.  29.  Ans.  $40.83|.  Ex.  80.  Ans.  $11.69+. 

Ex.  31.  Ans.  $13.88-1-.  Ex.  33.  Ans.  $2.43—. 

Ex.  34.  Ans..  $26.70+.  Ex.  39.  Ans.  $5019.44*. 

Ex.  41.  Time,  2  yr.  7  mo.  13  da.;  $2078.87—,  Ans. 

Ex.  42.  Time,  7  yr.  8  mo.  16  da.;  $4615.43+,  Ans. 
(302-308) 


150  PERCENTAGE. 

Ex.  43.  Time,  2  yr.  4  mo.  15  da.;  $42.86+,  Ans. 

Ex.  44.  21  yr.— 15  yr.  3  mo.  20  da.=5  yr.  8  MO.  10  da.; 
$3754.45+int.  for  5  yr.  8  mo.  10  da. 

=r$5037.22+,  Ans. 

Ex.  45.  He  gains  the  difference  between  the  legal  rates  of 
interest  in  the  two  places,  which  is  1  %  on  $7500  ; 
hence,  $7500X-01=$75,  Ans. 

Ex.  46.  Cash,  $3200 

$3500+int.  for  6    mo.=  3622.50 
$2500+  «    «   10  «    =  2645.83| 
$2120+  «     "   15  «    =  2305.50 

Ans.  $11773.83] 

Ex.  47.  $6840+int.  from  May  10, 1859,  to  June  21,  1860, 

.  =87297.14,  the  cost  of  the  flour  ; 
$6840-^-$5.70=1200  barrels  purchased; 
$6.625X1200=17950  receipts; 

$7950— $7297.14=8652.86,  Ans. 

Ex.  48. 

As  he  pays  7  %  per  year  of  360  days,  and,  by  agreement, 

receives  6  %  per  year  of  365  days, 

The  int.  in  N.  Y.  =||§  =  T783o  of  a  year's  int., 

And  the  int.  inO.=:||f  =  |  of  a  year's  int. 

Hence,  he  pays  $15000  x  T5<j  X  -^  =  $425.83^, 

He  receives  $15000        x  r{fo  x  f     =$360.00, 

And  he  loses  $65.8.3 1.  Ans. 

Ex  49.  $1250+  interest  for  7  mo.  16  da.  =  $1304.93 

$3540.84+  "         "  4    "     27    "  =  3642.049 

$  575+       «         «  3    «  =     585.062 

$816.90+  «         "  1    "      7    "  =     822.777 


ins.  $6354.818- 
(308,  309) 


SIMPLE   INTEREST.  151 

Ex,  50.  33x11=363  da.,  whole  time  for  which.he  rec'd  int.; 
.363-r-6=$.0605,  interest  on  $1  at  6  % ; 
$.0605X4=$.242,      "       «    "    «2%amonth; 
$.242— $.06=1.182,  interest  gained  on  $1; 
$.182X1000=1182,  Ans. 

Ex.  51.  He  gained  at  the  rate  of  12|  %—  7f  %=4f  % 
per  annum,  for  2  yr.  5  mo.  10  da.=2|  jr.;  hence 
$21840 X.04f  X2|=$2535.86|,  Ans. 

Ex.  52.  $27.50X450=$12375,  money  borrowed; 

$12375+  int.  for  4  yr.  11  mo.  20  da. =$15759.22, 
outlay;  $34      X 180 =$6120 

$32.55X270=  8788.50  returns,  14908.50; 


Ans.,  $850.72. 

Ex.  53.  $1.12^X4500=15062.50,  cost; 

$1.06X4500  =  $4770,  wheat  sold  for; 
$4770 X- 05  =  $238. 50,  interest; 
$4770_f-$238.50=$5008.50,  total  returns; 
$5062.50— $5008.50=$54,  loss,  Ans. 


PAETIAL    PAYMENTS. 

(543,  page  312.) 
Ex.2. 

Amount  of  note,  Oct.  15,  1860,  (1  yr.),     $1272.00 
Payment,  1000.00 


New  principal,  $272.00 

Amount,  April  15,  1861,  (6  mo.),  280.16 

Payment,  200.00 


New  principal,  «  $80.16 

Amount,  Oct.  15,  1861,  (6  mo.),  82.56+,  Ans. 

(309—312) 


152  PERCENTAGE, 

Ex.3. 

Amt.  of  note,  Mar.  4,1856,  (8  mo.  24  da.),  $888.19+ 

Payment,                                              .  210.93 

New  principal,  $667.26+ 

Am't,  July  9,  1857,  (1  yr.  4  mo.  5  da.),  732.01+ 

Payment,  140.00 

New  principal,  $592.01+ 

Amount,  Feb.  20,  1858,  (7  mo.  11  da.),  613.81+ 

Payment,  178.00 

New  principal,  $435.81+ 

Am't,  May  5,  1859,  (1  yr.  2  mo.  15  da.),  467.41+ 

Payment,  154.30 

New  principal,  $313.11+ 

Amount,  Jan.  17, 1860,  (8  mo.  12  da.),  326.26+ 

Payment,  259.45 

New  principal,  $66.81 

Am't,  Oct.  24.  1861,  (1  yr.  9  mo.  7  da.),  73.90+,  Ans. 

Ex.4. 

Amt.  note,  Oct.  20, 1861,  (1  yr.  1  mo.  16  da.),  $415,34+ 

Payment,  126.50 

New  principal,  $288.84 

Amount,  Jan.  1,  1862,  (2  mo.  11  da.),  292.83 — ,  Ans. 

Ex.5. 

Amt.  of  note,  June  1, 1857,  (2  mo.  25  da.),  $3516.02+ 

Payment,  1247.60 

New  principal,  $2268.42+ 

Amount,  Sept.  10, 1857,  (3  mo.  9  da.),  2299.61+ 

Payment,                                          *  1400.00 

(312) 


PARTIAL    PAYMENTS.  153 

•» 

New  principal,  $899.61+ 

Amount,  Jan.  31,  1858,  (4  mo.  21  da.),  917.23+,  Ana. 

Ex.6. 

Mortgage,                                                  -  $9750.00 

Payment,  Oct.  1,  1860,  846.50 

Due,  Jan.  1,  1861,  (9  mo.  from  date),  $8903.50 

Amt.,  Oct.  20, 1862,  (1  yr.  9  mo.  19  da.),  10508.60+ 

Payment,  2500.00 

New  principal,  $8008.60+ 

Amount,  July  3, 1863,  (8  mo.  13  da.),  8571.43— 

Payment,  1500.00 

New  principal,  $7071.43 — 

Amount,  Jan.  1,  1864,  (5  mo.  28  da.),  7421.06  + 

Payment,  500.00 

New  principal,  $6921.06  + 

Amount,  Apr.  1,  1865,  (1  yr.  3  mo.),  7786.19+,  Ans. 

Ex.7. 

Amount  of  note,  May  1,  1861,  (3  mo.),  $507.50 

Payment,  40.00 

New  principal,  $467.50 

Amount,  May  1,  1862,  (1  yr.),  495.55 

Sum  of  payments,  $8+$12+$30=  50.00 

New  principal,  $445.55 

Amt.,  Sept.  16, 1862,  (4  mo.  15  da.),  455.57+,  Am. 

(546,  page  314.) 
Ex.  1. 

Amount,  Apr.  1, 1857,  (1  yr,  2  mo.),  $1070 

1st  payment,  80 


(312-314) 


154 


PTOCENTAGE. 


New  principal, 

Amt.,  Apr.  1,  1858,  (1  yr.), 

Amt.  of  2d  payment   *)  date,  (8  mo.)? 

New  principal, 

Amt.,  Dec.  1,  1858,  (8  mo.), 

Sum  of  3d  and  J'Ji  payments, 


$  990 
1049.40 
31.20 

$1018.20 
1058.93- 
610.00 


266.38—,  Ans. 


New  principM,  $448.93 — 

Amt.,  Oct.  1,  1859,  (10  mo.),  471.38— 

Amt.  of  WJL  payment,  to  date,  (5  mo.),  205.00 

Ex.  2. 

Amt.  of  principal,  Jan.  10, 1863,  (4  yr.  8  mo.),  $2560 

Amt.  of  fet  indorsement,  (3  yr.  10  mo.), 
"     "    2d  "          (2  yr.  8  mo.), 

"     "   3d  "          (1  yr.  4  mo.), 


Ex.3. 

Amount,  May  10,  1859,  (1  yr.), 
Am't  of  1st  payment  from  Mar.  10  to  May 
10,  1859,  (2  mo.), 

New  principal, 

Ain't,  May  10,  1860,  (1  yr.), 

2d  payment, 

New  principal, 

Am't,  Sept.  10,  1861,  (1  yr.  4  mo.), 

3d  payment, 

New  principal, 

Ain't,  Jan.  10,  1863,  (1  yr.  4  mo.), 
(314) 


808 

$1312 
1390.72 
400.00 

$990.72 
1069.98+ 
300.00 

$769.98+ 
831.58+,  Ans. 


PAKTIAL   PAYMENTS.  156 

By  the  New  Hampshire  Rule. 

Ex.4. 

Am't  of  note,  May  10,  1859,  (1  yr.),  $2120 

Am't  of  1st  payment,  to  date,  (2  mo.);  808 

New  principal,  $1312 

Am't,  May  10,  1860,  (1  yr.),  1390.72 

2d  payment,  400 

New  principal,  $990.72 

Am't,  May  10,  1861,  (1  yr.),  1050.16+ 

Am't,  May  10,  1862,  (1  yr.),  1113.17+ 

Am't  of  3d  payment,  (8  mo.),  312.00 

New  principal,  $801.17+ 

Am't,  Jan.  10,  1863,  (8  mo.)  833.21+,  Am. 

By  the   United  States  Rule. 

Am't,  Mar.  10,  1859,  (10  mo.),  $2100 

-  Payment,  800 


New  principal,  $1300 

Ain't,  May  10,  1860,  (1  yr.  2  mo.),  1391 

Payment,  400 


New  principal, 

Am't,  Sept.  10,  1861,  (1  yr.  4  mo.),  1070.28 

Payment,  .         300 

New  principal,  $770.28 

Am't,  Jan.  10,  1863,  (1  yr.  4  mo.),  831.90+,  An* 

(314) 


166  PERCENTAGE. 

SAVINGS   BANK  ACCOUNTS* 

(549,  page  316.; 

Ex.2. 

Sum  of  deposits,  Jan.  20,  $26.80 

Draft,  Jan.  28,  5 

Balance,  Feb.  1,  $21.80 

Draft,  Feb.  7,  8.48 

Least  balance  during  current  month,  $13.32 

Sum  of  deposits,  Feb.  20  and  27,  24.85 

Int.  on  $13.32  for  1  mo.,  .07 

Balance,  Mar.  1,  $38.24 

Deposit,  Mar.  6,  14.65 

$52.89 

Draft,  Mar.  20,  10 

$42.89 

Deposit  Mar.  29,  7.98 

Int.  on  $38.24,  (least  bal.)  for  1  mo.,  .19 

Balance,  Apr.  1,  $51.06 

Draft,  April  11,  12.76 

$38.30 

Deposit,  April  25,  3.49 

Int.  on  $38.30  (leas^bal.)  for  1  mo.,  .19 

Balance,  May  1,  $41.98 

Sum  of  deposits  during  May,  72.29 

Int.  on  $41.98,  (least  bal.)  for  1  mo.,  .21 

(316) 


SAVINGS   BANK   ACCOUNTS.  157 

Balance,  June  1,  $114.48 

Sum  of  drafts  during  June,  31.92 


$82.56 
Int.  on  $82.56  (least  bal.)  for  1  mo.,  .41 


Balance,  July  1,  $82.97 

Deposit,  July  28,  15.68 

Int.  on  $82.97  (least  bal.)  for  1  mo.,  .41 

Balance,  Aug.  1,  $99.06 

Deposit,  Aug.  3,  18.45 

$117.51 

Draft,  Aug.  17,  5.64 

$111.87 

Deposit,  Aug.  26,  4.50 

[nt.  on  $99.06  (least  bal.)  for  1  mo.,  .50 

Balance,  Sept.  1,  1860,  $116.87,  AHS. 

Ex.3. 

Balance,  Jan.  1,  1859,  $  47.50 

Deposit,  March  12,  1859,  124.36 

Int.  on  $47.50  for  3  mo.,  .83 

Balance,  April  1,  1859.,  $172.69 

Draft,  May  12,  1859,  50  36 

$122.33 

Deposit,  June  20,  1859,  130.56 

Int.  on  $122.33  (least  bal.)  for  3  mo.,  2.14 


Balance,  July  1,  1859,  $255.03 

Deposit,  Aug.  3,  1859,  68.75 

Int.  on  $255.03  (least  bal.)  for  3  mo.,  4.46 

(316) 

14 


158  PERCENTAGE. 

Balance,  Oct.  1, 1859,  $328.24 

Sum  of  drafts,  Oct.,  Nov.,  and  Dec.,  1859,  74.76 

$253.48 
Int.  on  $253.48  (least  bal.)  for  3  mo.,  4.43 


Balance,  Jan.  1,  I860,  $257.91 

Deposit,  Jan.  25,  1860,  160.80 

Int.  on  $257.91  (least  bal.)  for  3  mo.,  4.51 

Balance,  April  1  or  4,  1860,  $423.22,  Ans. 
Ex.4. 

Balance,  July  1,  1858,  $175 

Draft,  Sept.  14,  1858,  65 


Deposit,  Nov.  1,  1858,  150 

Int.  on  $110  (least  bal.)  for  6  mo.,  3.30 

Balance,  Jan.  1,  1859,  $263.30 

Deposit,  Feb.  24,  1859,  200 

Int.  on  $263.30  (least  bal.)  for  6  mo.,  7.90 

Balance,  July  1,  1859,  $471.20 

Draft,  July  25,  1859,  120 

$351.20 
Deposit,  Sept.  10,  1859,  56 

$407.20 
Draft,  Dec.  3,  1859,  80 

$327.20 
Int.  on  $327.20  (least  bal.)  for  6  mo.,  9.82 

$337.02,  Am. 
(316,  317) 


SAVINGS    BANK  ACCOUNTS.  159 

Ex.  5. 

Deposit,  Jan.  ],  1858,  $36.50 

Int.  on  the  same  for  6  mo.,  .91 

Deposit,  Mar.  17,  1858,  25.38 

Int.  on  the  same  for  3  mo.,  .32 


Balance,  July  1,  1858,  $63.11 
Deposit,  Aug.  1,  1858,               $84.72 
Draft,  Sept.  16,  1858,                  36.16 

48.56 

Int.  on  $63.6Q  for  6  mo.,  1.58 

"     "  $48.56   "  3   "  .61 


Balance,  Jan.  1,  1859,  $113.86 

Draft,  Jan.  27,  1859,  $13.48 

«     Mar.  1,       «  17.50 

30.98 


Deposit,  June  11,  1859,  50 

Int.  on  $84.26  (least  bal.)  for  6  mo.,  2.07 

Balance,  July  1,  1859,  $134.95 

Deposit,  Nov.  16,  1859,  40.78 

Int.  on  $137.21  (least  bal.)  for  6  mo.,  3.37 



Balance,  Jan.  1,  1860,  $179.10, 
(317) 


160  PERCENTAGE. 

COMPOUND  INTEKEST. 

(55O,  page  318.) 

Ex.  1.  $750      Principal  for  1st  year ; 

$750X1.06=          795  «        «   2d     « 

$795X1.06=          842.70       «        «   3d     « 
$842.70x1.06=      893.26       «        "   4th    " 
$893.26X1.06=      946.86,  Amount  "    4  years; 
750        Principal ; 


Ans.  $196.86 — ,  Compound  interest. 

Ex.  2.  $250       Principal  for  1st  year ; 

$250X1.07=  267.50  «  «  2d  " 
$267.50X1.07=  286.23  «  "  3d  « 
$286.23X1.07=  306.26+,  J.TIS. 

Ex.  3.  $1475.50  Principal  1st  half  yeai , 

$1475.50xl-035=$1527.15  "  2d  « 
$1527.15X1.035=  1580.60  «  3d  « 
$1580.60X1.035=  1635.92  «  4th  « 
$1635.92X1.035=  1693.18  «  5th  « 
$1693.18X1.035=  1752.43+,  Ans. 

Ex.  4.  $376      Principal  for  1st  year; 

$376X1.06=       398.56      «         "   2d      « 
$398.56X1.06=       422.47     .«         «  3d      « 
$422.47X1.06—       447.81  Prin.  for  8  mo.  15  da.; 
$447.81X1.0425=  466.84  Amt.  3  yr.  8  mo.  15  da.; 
376 

$  90.84+,  Ans. 

(55 1 5  page  320.) 

Ex.  2.    $536.75X2.51817=$1351.63— ,  Ans. 
(318-320) 


PROBLEMS  IN  INTEREST.          161 

Ex.  3.     There  will  be  5  half  years  and  3  mo.  12  days  over ; 
$.019|,  simple  interest  of  $1  for  3  mo.  12  da.; 
$.19^|       "  "         "    "     "  2  yr.  9  mo.  12  da.; 

$1275X1.187686=81514.299+,  amt.  of  the  given 
sum  at  3^  %  per  half  year,  for  2  yr.  6  mo. ; 
$1514.299xl.019f =$1544.332+,  amt.  for  2  yr. 
9  mo.  12  da.; 
$1544.331—1  .194f =$1292.51— ,  Am. 

Ex.  4.     There  will  be  7  interest  terms  of  3  mo.  each,  and  1 
partial  term  of  1  mo.  20  da. ;  hence, 
$1840x(1.02)7Xl.OH=$2137.06+,  Ans. 

Ex.  5.     21  yr.— 12  yr.  7  mo.  12  da.  =  8  yr.  4  mo.  18  da.; 
$15000X1.733986=126009.79,  amt.  for  8  yr.; 

697.93,  int.  for  4  mo.  18  da.; 

$26707.72,  Ans. 

Ex.  6      Since  $1  will  amount -to  $3.869685,  the  sum  which 
will  amount  to  $2902.263  in  the  same  time  is 
$2902.263 -f- 3.869685 =$750,  Ans. 


PROBLEMS  IN   INTEREST. 

(552,  page  321.) 

Ex.  1.     Int.  of  $1  for  1  yr.  6  mo.  is  4.0975 ; 

$279.S25-^.0975=$2870,  Ans. 

Ex.  2.     Int.  of  $1  for  6  mo.  24  da.  is  $.0425; 
$63.75-5-.0425=$1500,  Ans. 

Ex.  3.     Int.  of  $1  for  10  da.  is  $.002|=$^fT 

$12^4^ =$4500,  Ans. 

Ex.  4,     $3125-?-.125=$25000,  Ans. 
(320,  321) 


162  PERCENTAGE. 

Ex.  5,     $30x12  =  1360,  yearly  income  ; 
$360-r-.095=$3789.47+,  Ans. 

Ex.  6.     Int.  of  $1  for  6  yr.  5  mo.  11  da.  is  $.451$  J; 
$3159.14-=-.451|J=$7000,  Ans. 

Ex.  7.     The  compound  interest  of  $1  for  2  yr.  9  mo.  is 
$.174162;  $69.67--.174162=$400+,  Ans. 

Ex.  S.     The  compound  interest  of  $1  for  1  yr.  6  mo.  15  da.  is 
$.09445;  $124.1624--.09445  =  $1314.583+,  Ans. 

(553,  page  322.) 

Ex.  1.     The  amount  of  $1  for  2  yr.  3  mo.  10  da.  is  $1.113| ; 
'  $1893.61$-f-1.113f =$1700,  Ans. 

Ex.  2.     The  amount  of  $1  for  3  yr.  5  mo.  12  da.  is  $1.207 ; 
$681.448-f-1.207=$564.58— ,  Ans. 

Ex.  3.     The  amount  of  $1  for  10  yr.  2  mo.  is  $1.35^; 
$15660-f4.35T\=$11550.09+,  Ans. 

Ex.  4.     The  amount  of  $1  for  2  yr.  8  mo.  29  da. 
is  $1.1923055+; 

$1568.97-7-1.1923055=81315.913— ,sum  at  int.; 
$1568.97— $1315.913=$253.057+,^7is. 

Ex.  5.     The  amount  of  $1  for  243  da.  is  $1.054; 

$11119.70-^054=110550,  sum  at  interest; 
$11119.70— $10550=$569.70,  Ans. 

Ex.  6.     The  amount  of  $1  at  compound  interest  for  4  yr.  is 
$1.262477 ;  $8644.62-=-1.262477=$6847.34+. 

Ex.  7.     Amount  of  $1  for  10  yr.  5  mo.  is  $1.835619 ; 
$26772.96-f-1.835619=$14585.24+,  Ans. 

(321,  322) 


PROBLEMS   IN   INTEREST.  163 

(554,  page  323.) 

Ex.  1.     $9.375,  interest  on  $750  at  1  %  ; 

$796.876— $750=846.875,  whole  interest; 
$46.875--$9.375=5  %,  Am. 

Ex.  2.     $1.32f ,  interest  on  $1700  at  1  %  ', 

$10.58--$i.32f =8+  %,  Ans. 

• 

Ex.  3.     $57,  interest  on  $600  at  1  %  ; 

$856.50— $600=$256.50,  whole  interest; 
$256.50--$57=:4^  %,  Ans. 

Ex.  4.     $72.6628,  income  at  1  %  ; 

$744.7937-4-$72.6628=10;i  %,  Ans. 

Ex.  5.     By  investing  $100 — $30 =$70,  he  will  receive  an- 
nually $4+$4=$8;  and  $8-i-$70  =  llf  %,  Ans. 

Ex.  6.     Since  $100  at  1  %  will  gain  $4,  $6,  $8,  and  $10,  in 
4,  6,  8  and  10  years  respectively,   to  gain  $100  will 
require  100-i-  4=25     %,  for    4  years; 
100-f-  6=16f  %,   «     6  "  « 
100-4-  8=12,i  o/0j   «     g     " 
100-4-10=10    %,   «  10     " 

Ex.  7.     Since  to  triple  itself  the  interest  must  be  twice  the 
principal,  any  sum,  as  $100,  will  triple  itself  in 

2  years,  at  2JO^-  2  =  100  %  ; 

5     «       t(  200-r-  5=40     %; 

7  a  «  200-*-  7=284  %• 
12  «  «  200-4-12=16|  %; 
20  «  «  200-^-20  =  10  %.  • 

Ex.  8,     $760.50-4-$7800=9|  %,  Ans. 
Ex.  9,     $223X12 =$2676,  yearly  income; 

$2676-=-$35680=7^  %, 

(323) 


164  PERCENTAGE. 

(555,  page  324.) 

Ex.  1.     $273.51  X- 07  =  $19.1457,  interest  for  1  yr.; 

$312.864— $273.51  =  $39.354,  whole  interest; 

$39.354--$19.1457=2.0555+  yr.  =2  yr.  20  da. 
Ex.  2.     $650.82 X- 05  =  $32.541,  interest  for  1  yr.; 

$761.44— $650.82=$110.62,  whole  interest; 

$110.62--$32.541=3.3994+  yr. 

=3  yr.  4  mo.  24  da.,  Ans. 

Ex.  3.     Since  $100  at  3  %,  4^  %,  6  %,  7  %  and  10  %, 

will  gain  $3,  $4^,  $6,  $7  and  $10,  respectively,  in  1 

year,  a  sum  will  double  itself  in 

100-f-3=33|  yr.  at  3  %  }  100-4-4^= 22|  yr.  at  44 

%  ;  100--6=16f  yr.  at  6  %  ;  100V7=14f  yr.  at 

7  o/0  •  100-^10  =  10  yr.  at  10  %. 

And  quadruple  itself  in  300-^-3=100  yr.  at  3  % ; 

300-^-4^ =66 1  yr.  at  4|  %;  300-^-6=50  yr.  at  6  %; 

300~7=42f  yr.  at  7  %  ',.  300-^-10=30  yr.  at  10  %. 
Ex.  4.     $9750X-02  =  $195,  interest  for  1  month; 

$780-7-$195=4  mo.,  Ans. 

Ex.  6.  $376.76-f-333=$1.131414+  amt.  of  $1;  and,  re- 
ferring to  the  table  under  2^  %,  we  find  that  $1  will 
amount  to  this  sum  (very  nearly)  in  5  terms  of  in- 
terest, or  2^  years,  Ans. 

Ex.  7.  We  find  by  the  table  that  it  will  require  11  years  for 
$1  at  6  %  to  gain  $.898299;  and  $1 — $.898299 
=$.101701,  the  interest  still  to  be  gained.  $1.898299 
X- 06=$. 113898,  int.  on  last  amount  for  1  year. 
$,101701-:-$.113898=.8929  yr.  =  10  mo.  21+da. 
Hence,  any  sum  will  double  itself  at  6  %  compound 
interest,  in  11  yr.  10  mo.  21  da. 
Again,  $1  at  7  %  compound  interest  will  gain 


DISCOUNT.  165 

$.967151  in  10  yr.;  and  $1— $.967151 =$.032819, 
the   interest   still   to   be   gained;    $1.967151  X-07 
=$.137701  int.,  of  last  amount  for  1  year;  $.032849 
-4-$.137701=.23858  yr.=2  mo.  26— da. 
Hence,  10  yr.  2  mo.  26  da.,  Am. 


DISCOUNT. 

(557,  page  325.) 

Ex.  1.     $1.0275,  amount  of  $1  for  5  mo.  15  da. ; 
$385.3125--1.0275=$375,  Ans. 

Ex.  2.     $1.0826527+,  amount  of  $1  for  1  yr.  6  mo.  1  da.; 
$429.986--1.0826527  =  $397.160—,  present  worth; 
$429.986— $397.16=$32.826,  Ans. 

Ex.  3.     $1.12,  amount  of  $1  for  1  yr.  6  mo. ; 

$3665.20-4-1.12  =  $3272.50,  cash  value  of  sale; 

$3272.50— $2964.12=$308.38,  Ans. 

Ex.4.     $12000--1.05       =     $11428.57+ 
$15000-^1.125     =     $13333.33+ 

Latter  offer,  $24761.90+,  cash  value ; 

$25000— $24761.90=8238.10  gained,  Ans. 

Ex.  5.     From  Apr.  1,  I860,  to  Sept.  4, 1860,  is  5  mo.  3  da.; 

"         «     "      "      «  Jan.  1,  1861,  «  9  mo. ; 
Discounting  $1470  for  5  mo.  3  da.  at  10  %,  and 
$2816.80  for  9  mo.  at  10  %,  we  have 
$1470-r-1.0425=$1410.07+  borrowed,  payable 
Sept.  4,  1860;  $2816.80--1.075=$2620.28+, 
borrowed,  payable  Jan.  1,  1861 ; 

(324,  325) 


166  PEKCENTAGE. 

$1410.07+2620.28  =  $4030.35,  whole  loan; 
Ain't  of  $1410.07  @  7  %,  Sept.  4,  1860,  $1452.02 
«      «     2620.28  «      «      Jan.  1,  1861,     2757.84 


Money  borrowed,  with  interest,  $4209.86 

$1470+$2816.80=$4286.80,  money  to  be  paid,  if 
the  house  and  lot  were  bought  on  credit;  hence, 
$4286.80— $4209.86=$76.94,  gain  by  borrowing 
money  to  pay  down. 

Ex.  6     $576-=-1.08=$533.33| 

$576— $533.33|=$42.66f,  discount; 
$576 X- 08  =$46.08  ,  interest; 

Difference,  $3.41|,  Ans. 

Ex.  7.     The  term  of  discount  for  the  first  sum  is  6  mo.  25 
da.,  and  for  the  second,  11  mo.  14  da.;  hence, 
$243.16-f-1.039861  =  $233.843+ 

$178.64-^-1.066888=  167.444+ 


$401.287+,  Ans. 
Ex.  8.     $.09X488X120  =  $5270.40,  cost; 

$5270.40-r-1.06=$4972.07+,pres.  worth  of  note; 
$6441.60— $4972.07  =  $1469.53— ,  Am. 

Ex.  9.     $6.25-^-1.04=$6.009+,  present  worth  of  $6.25; 
$6.50-5-1.06=  6.132+,       «  «      «  $6.50; 


Ans.  $.123 ±,  difference  in  favor  of  $6.25 

Ex.  10.  5  %  Per  annum  is  3|   %  for  8  mo.;  hence 
$6400 X-03| =$213.33| ,  discount; 
$6400— $213.33 J=$6186.66§,  to  be  hired; 
$6186.66f  X-03|  =  $206.22f,  int.  on  hired  money 

$213.33}- $206.22|  =  $7.1H,  Ans. 

(325,  326) 


BANKING.  167 

BANKING. 

(576,  page  329.) 

Ex.  1.     Int.  of  $1487  at  6  %  for  33  da.=$8.18,  discount; 
81487—88.18=11478.82,  proceeds,  Ans. 

Ex.  2.     Int.  of  $384.50  at  7  %  for  93  da.— $6.95,  discount; 
$384.50— $6.95=$377.55,  proceeds,  Ans. 

Ex.  3.     Int.  of  $975  at  8  %  for  63  da.=$13.65,  discount; 
$975— $13.65=1961.35,  proceeds  of  note; 
$1000— $961.35=138.65,  Ans. 

Ex.  4.     195  A.  2  E.  25  P.=195;65625  A.; 

$27.50X195.65625=$5380.547+,  face  of  note; 

int.  of  $5380.547  at  7  %  for  4  mo.  18  da. 

=  $144.378,  discouDt; 

$5380.547— $144.378 =$5236.169,  Ans, 
Ex.  5.     From  Aug.  26  to  Nov.   29  is  95  da.,  the  term  of 

discount.     Int.  of  $1962.45  at  7   %  for  95  da.= 

$36.25 ;  $1962.45— $36.25=$1926.20,  proceeds. 
Ex.  6.     Int.  of  $1066.75  at  6  %  for  74  da.  =  $13.16,  dise't; 

$1066.75— $13.16=$1053.59,  proceeds. 
Ex.  7.     Note  due  Aug.  2  °/2 , ;  term  of  discount,  49  da. ; 

int.  of  $784.72  at  8  %  for  49  da.=$8.54,  discount, 

$784.72— $8.54=$776.18,  proceeds. 
Ex.  8.     Int.  of  $1845.50  at  24  %  for  31  da.=$38.14,  disc't; 

$1845.50— $38.14=81807.36,  proceeds. 

Ex.  9.     $950-f-  1.0175=$933.66,  true  present  worth; 
$950— $933. 66 =$16.34,  true  discount; 
$950 X- 0175     =   16.63,  bank  discount; 

$     .29,  Ans. 
(329,  330) 


168  PERCENTAGE. 

Ex.  10.  $1375.50-r~1.01=$1361.88,  true  present  worth; 
$1375.50— $1361.88=113.62,  true  discount; 
$1375.50X-01  =  13.76,  bank      " 


$  .14,  Am. 

(577,  page  330.) 

Ex.  1.     $1275-i-.9895=$1288.53— ,  Ans. 
Ex.  2.     $5000-f-.9845=$5078.72+,  Ans. 

Ex.  3.     The  proceeds  of  $1  for  3  mo.  at  7  %  are  $.9825 ; 
$276.S4-f-.9825=$281.77+,  Ans. 

Ex.  4,     The  proceeds  of  $1  for  4  mo.  3  da.  at  7|  %  are  $.974375 
$1486.90--.974375=$1526+,  Am. 

Ex.  5.     Proceeds  of  $1  for  6  mo.  3  da.  at  24  %  =  $.878; 
$496--.878= $564.92,  Ans. 

Ex.  6.     Proceeds  of  $1  for  33  da.  at  5  %  =$.9954166+ , 
$1200-5-.9954166=$1205.52+,  Am. 

Ex.  7.     Proceeds  of  $1  for  63  da.  at  18  %  =  $.9685; 
$575=.9685=$593.70,  Am. 

Ex.  8.     Proceeds  of  $1  for  110  da.  at  12  %=$.963| ; 
$187.50-r-.963| =$194.63+ ,  Am. 

(5785  page  331.) 

Ex.  1.     Discount  of  $100  for  33  da. =$.55 ; 
$100—1.55=899.45,  proceeds; 
interest  on  proceeds  at  1  %  =  $.0911625; 
$.55-^.0911625=6^3  %,  Ans. 

Ex.  2.     Discount  of  $100  for  2  mo.  3  da. =$4.20; 
$100— $4.20=$95.80,  proceeds; 
interest  on  proceeds  at  1  %  =  $.16765; 
$4.20-5-8.16765=25^  %,  Am. 
(330-332) 


BANKING.  169 

Ex.  3      Discount  of  $100  for  93  da.=$4.65; 

$100—  $4.65  —  $95.35,  proceeds; 

interest  on  proceeds  at  1  %  =  $.24632^; 

$4.65-Mfr.24632T'2=18if$4  %?  Am. 
Ex.  4.     Am.,  5 
Ex.  5. 


(579,  page  332.) 

Ex.  1. 

$.45|=int.  of  $100  at  5  %  for  33  da.; 
8.55  =  "    "      "         6  %     "      " 
$.64|=  a    a      «         7  ^     a      « 
$.91f  —  «    «      «       10  %     "      « 

100.45|X.0045| 

$  -  =int.  at  1  %  of  amt.  of  $100  at  5  %  \ 
5 

100.55  X.  0055 


6 
100.641  X. 00641 


7%; 


7 

100.91|  X-0091f 
$  -  =  «    «     «  «  «       10  % 

Hence 

5  5 

.45  JX  -  =:  -  =4§|ff  %    cor.  to  5 
100.45|  X.0045|     1.0045| 

6  6 

.55  X  ---  =  -  =^H%      "      "  6 
100.55X.0055          1.0055 

7  7 


100.64  JX.  0064J     1.0064  J 

(332) 
15 


170  PERCENTAGE. 

10  10 

,91|  x  --  =  -  =9TWr%  cor.  to   10  %] 
100.91|  X  -0091|     l.OOOlf 

NOTE.  —  From  the  solution  given  above,  we  see  that  the  result, 
after  cancellation,  is  obtained,  by  dividing  the  given  rate  by  the 
amount  of  $1  for  the  given  time. 

Ex.  2.     $1.0206f  =amt.  of  $1  at  8  %  for  3  mo.  3  da.; 

.08H-1.0206f  =  7{fff  #,  Ans. 
Ex.  3.     1     %  per  mo.  =12  %  per  annum; 

1^         "     "  =18          "       " 

2  "     "  =24:          "      " 

2%         "     "  =30          "       " 

$1.0210=:amt.  of  $1  for  63  da.  at  12  %  , 
m  $1.0315=    "          «  «      -<  18  %  , 

$1.0420=    "          "  "      «  24  %  ; 

$1.0525=    «  «  «      "  BO  %  ; 

.12-^-1.021   =ll-iVA  %>  to  PaJ  !   %  Per  m<>.  j 


.24-V-1.042  =28^27T     %,  "    «    2  %  « 
.30-j-1.0525=28|if    %,  "    «    2^%  " 
Ex.  4.     81.105=anit.  of  $1  for  18  mo.  at  7  %  ; 
.07-^-1.105=6^  ^  Ans- 


EXCHANGE. 

(S99,  page  336.) 

Ex.  1,     $1.005— $.0064I  =  $.9985|,  cost  of  exchange  for  $1 ; 
$5400X.9985|=$5392.35+,  Ans. 

Ex,  2.     $1 — $.0325=8.9675,  cost  of  exchange  for  $1 ; 
J3000X-9675=$2902.50,  Ans. 

(332-336) 


EXCHANGE.  171 

Ex.  3.     $1.01375— $.0155=$.99825,  cost  exchange  for  $1 ; 
$4800  X- 99825=84791.60,  Am. 

Ex.  4.     $550.62-s-1.035=$532,  Am. 

Ex.  5.     $1.0175— 8.0064  J  =  $1.0110$,  costexcha,nge  for  $1; 
$J324.74-r-$1.0110f =$1310.22— ,  Ans. 

Ex.  6.     $7500 X- 015=8112.50,  commission; 

17500— $112.50 =$7387.50,  net  proceeds  of  sale ; 
87387.50-5-1.005=$7350.75— ,  Ans. 

Ex.  7.     $4800 X- 005 =$24,  discount; 

D— $24=    $4776     rec'd  per  dft.  on  Baltimore ; 
3-5-1.0075=4764.27  "     "     "     "   Hartford; 


$11.73,  Ans. 

Ex.  8.     $508.75-5-1.0175=8500,  dividend ; 

$500-;-.0625=$8000=80  shares,  Ans. 

Ex.  9.     $5075— $5000=$75,  premium ; 

$75-v-$5000=.015=l^  %,  rate  of  premium,  Ans. 

Ex.10.  $5141.78-^-5320=    $.9665,  cost  of  exchange  for  $1; 
int.  of  $1  for  63  da.  =  .01225 


$.97875,  course  of  exchange; 
$1—1.97875 =$.02125 =2|  %,  rate  of  discount. 

(6O6,  page  342.) 

Ex.  1.    £325  3  s.  9  d.=£325.1875 ; 
40X1.0975 

£325.1875  X =$1586.19+,  Ans. 

9 

Ex.  2.     6000  X$.1875^$1125,  Ans. 

Ex.  3.     $.754xl.0125=$.763425,  course  of  exchange; 
3000  roubles X- 763425 =$2290.275,  Ans. 

(336-342) 


1 72  PERCENTAGE. 

Ex.  4.     $S31.12-*-.96=865.75  scudi 
=865  scudi  9  carlini,  Am. 

Ex.  5.     $40xL08=£9,  course  of  exchange; 
£9 

$1— : —  cost  of  a  unit  of  U.  S.  currency; 

40XL08 

9 

£125--: : —$600,  Ans. 

40  x  1.08 

40X1.11 

Ex.  6.     $7536.30- =£1527  12  s.  6J«  d.,  Ans. 

9 

Ex.  7.     9087  gilders  10  stivers =9087.5  gilders; 
9087.5-=-2.5=$3635,  Ans. 

Ex.  8.     1  milree=£|^  ;  £2500-j-|£^=9302.325f  f 
milrees— 9302  milrees  325||  reis,  Ans. 

Ex.  9.     i  1736--360=$4.82f  ?  cost  of  £1 ; 
$4.82f— $4.44f  =  $.37|,  premium; 
$.37^--$4.44|=r. 085=8^  %  premium,  Ans. 

Ex.  10.     1500--  .930 =$1612.90,  draft  on  Baltimore  j 
1500X1.065=81597.50,     «     «  Madeira; 

Difference,  $     15.40,  Ans. 

Ex.  11.  25256  lire  16  soldi=25256.8  lire; 

25256.8-r-131= 192.8  oz.  of  fine  gold  required; 
192.8  oz.x480=92544gr.     "       "          « 
92544--23.22=$3985.53,  value  of  gold  in  U.  S.  com, 
$3985.53 X«03=$119.57,  cost  of  exportation; 
$3985.53+$119.57=$4105.10,  cost  of  goods,  paid 
in  gold;  25256.8 X$.16=$4041.09,  cost  of  goods, 
paid  by  draft  on  Leghorn ; 
$4105.10— $4041.09=164.01,  Ans. 
(342,  343)      ' 


EXCHANGE.  173 

Ex.  12.  In  £  1  there  are  240  d.;  and  since  07  d.r=l  oz.  of 
pure  silver  240  d.  are  equal  to  240-f-67=3.58209 
+oz.= 1719.4032  gr.  of  fine  silver.  But  in  the 
coinage  of  1853, 1  half  dollar=192x. 9  =  172.8  gr 
of  fine  silver;  and  1  dollar=345.6  gr.;  hence 
1719.4032-r-345.6=$4.975,  Am. 

Ex.  13.  From  the  table  we  find  that  the  intrinsic  value  of  1 
thaler  is  $.692  in  the  coinage  of  1837 ;  hence 
$.692xl25=a:$86.50,  intrinsic  value  of  125  thalers; 
$88.23—886.50=81.73,  premium; 
8U3-7-886.50=.02=2  %,  Ans. 

(608,  page  346.) 

Ex.  1.     (?)  roubles=85000 
$40xl.lO=£9 

£1=6.48  roubles 
1.0025 


(?)=6610  roubles  74  copecks,  Ans. 

NOTE.  —  Since  proceeds,  or  the  value  of  the  money  in  the  place 
where  exchange  ends,  is  required,  we  place  1  plus  the  rate  of 
brokerage  on  the  left. 

Ex.  2.  (?)  $—7680  marcs 

5.16  francs=$l 

1  marc  =2    francs 


(?)=$3162.79,  Ans. 
Ex.  3.        (?)  C.=$750  N.  O. 

1  N.  0.^.9875  C.      * 


(?)= $740.625,  proceeds  by  direct  exchange; 
(?)  C.=$750  N.  0. 
1N.Y.=  1.015  C. 
IN.  0.=  .99  N.  Y. 
.995 


(343-346,) 


174  PERCENTAGE. 


(?)==$749.87;  proceeds  by  arbitration; 
740.63 


Ans.  $9.24+,  gained  by  arbitration. 

Ex.  4.  (?)$=6000  gilders 

11-1  gilders  =£1 


1.0125 


(?)=$2557.66+,  proceeds  by  arbitration; 
6000X-415=  2490      ,  "        "  direct  ex.; 

Ans.  $67.66,  gain  by  arbitration. 

Ex.  5.  (?)$=!  franc 

£9=$40Xl.09f 
26.86  francs  =£1 


(?)=$.181— ,  Ans. 
Ex.  6.     (?)  gilders=$6400 
$40xl.08=£9 

£1=240  d. 
18  d.  =  l  gilder 
1.005         .995 


(?)  =  17600  gilders  17+  stivers,  proceeds  arbitration 
$6400-^-.40  =  16000      "  •  "      direct  ex.; 


1600  gilders  17  stivers,  Ans. 

Ex.  7.     $5800 X- 95 =$55!!,  value  of  stock  in  N.  0. ; 
.     (?)  P.  =  $5510  N.  0. 
1  N.  Y.  =  1.02  P. 
1  N.  0.=  .995  N.  Y. 
1.0025 


(?)=$5606.08,  Ans. 
(346,  347) 


EQUATION   OF   PAYMENTS.  175 


Ex.  8.     (?)  £=$3000 
$1      =  5.40  f. 
185  f.=  100m. 
2m.=  35s. 

220s.  =  £1 


(?)  =    £696  lls.  2  d.,  proceeds  by  arbitration  ; 

63000X9 

-  =   613  12s.  9d.,        "     by  direct  exchange; 
40x1-10    -- 

£82  18s.  5d.,  gain  by  arbitration. 

EQUATION    OF   PAYMENTS. 

(617,  page  349.) 
Ex.  1. 

$500X30=$15000 

400X60=  24000  93000-f-1500=62  da.,  term  of  credit; 
600X90=  54000  Jan.  1+62  da.= 

Mar-  8>  1860'  e(luated  time- 


Ex  2.    1+|=T7,;  !-&=&; 

i  y  3=1 

i  X  8=2 

T52  X  12  =  5  8-5-1=8  mo.,  Ans. 

I  8 

Ex.  3.    $480X90=843200 
320X60=  19200 


$800          $62400 

62400-^-800^-78  da.,  average  term  of  credit; 
hence  the  note  for  the  whole  amount  will  run 
365  da.— 78  da.=287  da.     Again, 
(347-350) 


176 


PERCENTAGE. 


$480-4-1.015 =$472.906,  present  worth  of  $480  ; 
$320-1-  1.01=  316.831,     «  "       «     320; 

$789.737,  present  worth  both  debts ; 
Amount  of  $800  for  287  da. =$838.266 
"  $789.737  for  lyr.=  837.121 


$     1.14+,  AM. 
Ex.4. 

$280x3=$  840 

300x4=  1200    6400-^-1340 =4  mo.  23  da.,  average  Or.; 
200X5=  1000    Apr.  l-f4  mo.  23  da.=Aug.  24,  Ans. 
560x6=  3360 


$1340 


Ex.  1. 


$6400 


(618,  page  352.) 


Due. 

Da. 

Items. 

Prod. 

Mar.  4 
«  25 
Apr.  16 
"  30 
May  17 

0 
21 

43 
57 
74 

250 
960 
96 
200 
850 

7560 
4128 
11400 

25900 

48988 

1256 

48988—1256=39  da.; 
Mar.  4+39  da.= 
Apr.  12, 1860,  Ans. 


Ex.2. 


Due. 

Da. 

Items. 

Prod. 

Oct.   12,1859 
Nov.    6,  " 
Jan.  15,1860 
Mar.  19,    " 
Feb.  24,    " 
Mar.  25,    « 

0 

25 
95 
159 
135 
165 

ISO 
300 
150 
350 
130 
MO 

7500 

14250 
55650 
17f50 

mop 

K50 

118050 

Ex.3. 


Due. 

Da. 

Items. 

Prod. 

Nov.  30,  I860 

240 

Jan.  11,  1861 

42 

500 

21000 

Feb.  23,   « 
Mar.  17,   " 

Apr.  25,   " 

85 
10T 
146 

436 
325 
436 

37060 
347T5 
63656 

May  16,   " 

16T 

537 

89679 

2474 

246170 

118050-^-1250=94 
da.;  Oct.  12,  1859 
+94  da.= 
Jan.  14, 1860,  Ans. 

246170-^2474 
=100da.; 
Nov.  29,  1860+ 
100  da.= 
Mar.  10,  '61,  Am. 


(350-352) 


EQUATION   OF    PAYMENTS. 


177 


Ex.4. 


Due. 

Da. 

Items. 

Prod. 

Aug.  16,  1859 
Oct.  15,  " 
Dec.  14,  " 
Feb.  12,  1860 

60 
120 
180 

o50 
250 
300 

248 

15000 
36000 
44640 

1148 

95640 

9564Q-*-1148==88 

da.;  Aug.  16, 1859 
+83  da.=Nov.  7, 
1859,  Ans. 


(63O,  page  356.) 


3x.  2. 


Due. 

Da. 

Items. 

Prod. 

Due. 

Da. 

Items. 

Prod. 

Mar.  1 
Apr.  12 
July  16 
Sept.  14 

0 

42 
137 
197 

436 
548 
312 
536 

23016 
42744 
105592 

May  27 
"   9 
June  20 
Aug.  3 

87 
69 
111 
155 

400 

650 
200 
84 

34800 
44850 
22200 
13020 

Balances, 

1832 
1334 

171352 
114870 

56482^-49 

1334 

114870 

8  =  113  da.; 

498 

56482 

Mar.  1,  1860+113  da. = June  22,  I860,  due. 


Ex.3. 


Focal  date. 


Due. 

Da. 

Items. 

Int. 

Due. 

Da. 

Items. 

Int. 

Apr.      1 
"       12 
Mar.   16 
June  25 

85 
74 
101 
0 

54.36 
28.45 
95.75 
26.32 

.77 
.35 
1.61 

Apr.    1 
June  18 
'»    12 
"    20 

85 
7 
13 
5 

50.00 
3000 
125.00 
150.00 

.71 
03 

.27 
.13 

Ba 

204.88 

2.73 
1.14 

355.00 
204.88 

1.14 

ances, 

1.59 

150.12 

Int.  on  $150.12  for  1  da. =$.02502  ;  $1.59^-1.02502 
=64  da. ;  June  25,  1858+64  da. = Aug.  28,  1859. 


Ex.4, 


Due. 

Da. 

Item?. 

Int. 

*.58 
.51 
.24 

.08 

Due. 

Da. 

Items. 

Int. 

Jan.      1 
Feb.       1 
Mar.     17 
Apr.       1 

95 
64 
20 
5 

36.72 
48,25 
72.36 
98.48  ' 

Jan.   10 
"     21 
Mar.  23 
Apr.     6 

86 
75 
14 
0 

98.72 
25.84 
15.17 
8.96 

1.42 
-32 
.04 

1.78 
1.41 

talance  of  Items, 

255.81 
148.69 

107.12 

1.41 

148.69 

Balance  of  Int., 

.37 

Int.  of  $107.12  for  1  da.  =  .018 ; 
$.37-^,018=20  da.,  Ans. 

(352-358^ 


178 


PERCENTAGE. 


Ex.5, 


Due. 

Da. 

Items.  I      Prod. 

Due. 

Da. 

Items. 

Prod. 

Apr.   25 

24 

1000 

24003 

1  Apr.  1 
"  21 

0 

20 

SCO 

324 

6480 

Balances, 

1000 

884 

24(00 
6480 

L" 

116 

17520 

Ex.6. 


17520-5-116=151  da.;  Api   l.-f  151da.=Aug.  30. 

Due. 

Da. 

Items. 

Prod. 

Due. 

Da, 

Items. 

Prod. 

Aug.  12 
Oct.  15 

11 

75 

684 
468 

7524 
35100 

Aug.   1 

0 

839 

Balances, 

1152 
839 

42624 

839 

42624-^-313  =  136  da.  j 

313 

42624 

Aug.  l.-|-136  da.  — Dec.  15,  Am. 


Ex.      7. 


Due. 

Da. 

Items. 

Prod. 

.   Due. 

Da. 

Items. 

Prod. 

Mar.      1 
June     1 

Aug.      1 

0 
92 
153 

500 

800 
600 

73600 
91800 

Apr.    1 

31 

1000 

31000 

Balances, 

1900 
1000 

1^5400 
31000 

134400- 

1000 

31000 

-900=149 

900 

134400 

Mar.  1.+149  da.= July  28,  Ans. 


Ex  3. 


Due. 

Da. 

Items. 

Prod. 

Apr.  12 
Sept.  20 

0 
161 

500 
1000 

161000 

1500 

161000 

161000-s-  1500  = 
107  da.;  Apr.  12+ 
_  107  da.  =  July  28, 
the  average  maturity  of  the  two  debts.  Hence  to 
discharge  the  whole  obligation  by  two  equal  payments 
at  an  interval  of  60  da.,  these  payments  must  be 
made,  one  30  da.  before  July  28,  and  the  other  30 
da.  after  July  28;  and  we  have 

July  28—30  da.  =  June  28,  date  of  1st  payment; 
July  28+30  da. = Aug.  27,      «       2d        « 

(356,  357) 


EQUATION    OF    PAYMENTS. 


179 


Ex.9. 


Due. 

Da. 

Items. 

Prod. 

Duo. 

Da. 

Items. 

Prod. 

Sept.  12 
"     20 
"     30 
Oct.    5 
"      .6 
"     29 

0 
8 
18 
23 
34 
47 

530.84 
236.48 
739.C6 
273.44 
194.78 
536.42 

1892 
13312 
6289 
6623 
25212 

Sept.  14 
"      25 
Oct.      3 
,t        17 

Nov.    16 
"       24 

13 
21 
35 
65 
73 

436 
320 
'  560 
370 
840 
660 

872 
41CO 
11760 
l°.9fSO 
5:  POO 
40880 

2511.52 

53328 

308  > 
2511.fti? 

U5-22 

Balances,          574.48 1 


71894^-574.48  =  125  da.; 

Sept.  12,  1859+125  da.=Jan.  15,  1860, 


Ex.  10. 


Due. 

Da. 

Items. 

Prod. 

Due. 

Da. 

Item-. 

Prod. 

June    1 
July  16 
Aug.  12 
Sept.  12 
Nov.  18 

0 
45 
72 
103 
170 

500.78 
21  6.94 
843.75 
*  94.37 
856.48 

97063 
60750 
61220 
145602 

June  3 
July   1 
Nov.  1 

2 
30 
153 

500 
1000 
1500 

1000 
30000 
229500 

Balances, 

4988.32 
3000.00 

364635 
260500 

3000  1     260500 

1988.32 

104135 

104135-v-1988.32=:52  da.; 
June  l-j-52  da. = July  23,  Ans. 


ACCOUNT   SALES. 

(625,  page  358.) 

Ex.  1.     We  first  average  the  sales,  using  Apr.  1,  the  earliest 
maturity  of  the  en- 
tire  account,  for  a 
focal  date,  thus  : 

695115—13362.80  =  52  da.; 
Apr.  1.-J-52  da.  — May  23,  date  for  commission  and 
guaranty.  We  now  average  the  entire  account, 
using  the  same  focal  date  as  before,  and  putting 

(357,  358) 


Due. 

I>a.  from 
Apr.  1. 

Items 

Pro,1  . 

May     15 
"        5 

June    28 

44 
34 

88 

43R8.iV) 
5344.80 
£650.0 

192192 
181723 
o21200 

=  52  da.: 

13362.80 

695115 

180 


PERCENTAGE. 


Freight,    Primage,  Wharfage  and    Cartage,  as  one 
item;  and  also,  commission  and  guaranty. 


Due. 

Da.  1    Items. 

Prod. 

Due. 

Da. 

Items. 

Prod. 

Apr.     1 
June    3 
June  28 
May    23 

0 
63 
88 
52 

185.20 
3207.07 
37.68 
443.27 

202055 
ofel5 
2S050 

May  15 
"      5 
June  28 

44 
34 

88 

48680.00 

5344.80 
3t50.°0 

1921  0-2 
181723 
o-l-CO 

|  3873.32 

228410 

13362.80 
3873.32 

695115 

228  11  0 

466705—9489.48=49  da.;     Balances, 
ipr.  1+49  da.inMay  20,  proceeds  due. 

9489.48 

466705 

Ex.  2.     Equation  for  the  average  of  sales. 

424900^-6275= 


da.   Mayl 

+68  da.  ==  July  8,  date  for  com- 
mission; $175.48+$  56.25  + 
$8.37=  $240.10,  charges  made 
May  1,  for  freight,  etc. 

Storage  on  200  bbl.  5  wk.= storage  on  1  bbl.  1000  wk.; 


Due. 

Da.  from 
May  1. 

Items. 

Prod. 

June  3 
"  30 

July  29 
Aug.  6 

33 
60 

89 
97 

1250 
2275 
2450 
£00 

41250 
136500 
218050 
29100 

424900 

6275 

"  "  350  "  9  "  =  " 
"  "  400  "  13  "  =  " 
"  "  50  "  14  "  =  " 


"     "      3150    " 
"     "     5200     " 

"     "       7CO     " 


10050  wk. 

10050X.02=$201  storage,  due  Aug.  6; 
$6275  X- 035 = $219.63  commission,  due  July  8. 


Due. 

Da. 

Items. 

Prod. 

Due. 

Da. 

Items. 

Prod. 

- 

May  1 

Aug.  6 
JulyS 

0 

97 
68 

240.10 

201.00 
219.62 

19497 
14935 

June    3 
!•     30 

July  29 
Aug.    6 

33 
60 
89 
97 

1250.00 
2275.00 
2450.ro 
300.00 

41250 

136500 
2180fO 
29100 

390468--5 

660.72 

34422 

6275.00 
660.72 

5614.28 

421°00 
34432 

390  168 

614.28=              Balances, 

70  da.;    May  1+70  da.=July  10,  proceeds  due. 
(358) 


PARTNERSHIP. 
SETTLEMENT   OF   ACCOUNTS   CURRENT. 

(626,  page  360.) 


181 


Ex.1. 


Due. 

Da.  |  Items.    Int. 

Cashval.      Due. 

Da. 

Items.      Int. 

Cash  val. 

Jan.  12 
"     26 
Feb.  13 
Mar.  16 
Apr.  25 

140 
126 
108 
77 
37 

500.36+11.67 
260.48+  5.26 
400.00+  7.20 
750.00+  9.63 
200.00+  1.23 

512.03      Jan.    1 
255.74     Feb.    3 
407.20      Mar.  26 
759.63      Apr.  20 
201.23      May  12 

151 
118 
67 
42 
20 

636.72+13.51 
486.57+  9.57 
1260.78+14.08 
756.36+  6.29 
248  79+     .83 

550/23 
496.14 
1274.86 
761.65 
249.6? 

2135.83 

8332.50 

$3332.50— $2135.83=$1196.67,  Am. 


Ex.2. 


Due. 

Da. 

Items.    Int. 

Cash  val. 

Due. 

Da. 

Items  Int. 

Cash  value.   ! 

Sept.  3 
Jan.    2 
•<     21 
Feb  12 
Dec.  15 

119 
2 
21 
43 
16 

478.36+11.07 
256.37-     .10 
375.26--  1.53 
8000-     .67 
148.76+    .46 

489.43 
256.27 
373.73 
79.33 
149.22 

Sept.  17 
"    20 
Oct.      3 
Nov.  17 
Bee.  27 

105 
102 
89 
44 
4 

96.54+1.P7 
200.00+3.97 
325.00+5.62 
50.00+  .43 
84.00+  .07 

98.61 

£0o.97 
.330.62 
50.43 
84.07 

1347.98 

767.G>J 

$1347.98— $767.60=1580.38,  Am. 


PARTNERSHIP. 

(629,  page  361.) 
Ex.  I.     $6470+$3780-f$9860=$20110 

$20110:  $6470=17890:  (?)=$2538.453,  A'sshaio 
$20110:  83780=$7890:  (?)  =  $1483.053,B's     " 
$20110  :$9860=$7890:  (?)  =  $3868.493,  C's     " 

Ex.  2.     $1847.50— $739=$1108.50,  C  pays; 

73900  2     "R^a  frflpfinn  •     110850 — 3     PV 

TS4750  —  5?  -E      traction  ;    iH?fw— f*  ^8 

$375X|=$225,  C's     «  ' 

(360,  361) 

16 


182  PARTNERSHIP. 

Ex.  3.     $10000  £8£88  =  IHb  A's  fraction  ; 

12800  ii{HJ*  =  lt,B's       " 
3200 


$26000 

$9400—  ($1500+$3400)=$4500,  net  profits; 
$4500xff  =$1730.77,  A's  s]bare  of  net  profits; 
$4500  X  if  =  2215.38,  B's      "          "       « 
$4500x<^=     553.85,  C's      "          «       « 
$553'.85+$1500=$2053.85,  C;s  whole  income. 

Ex.  4.     115A.32P.  =  115.2  A.;  $3.75x115.2  =$432,  rent; 

144+160+192+324=820  slieep; 

820  :  144=$432  :  (?)  =  »75.86,  A  pays; 

820  :  160  =  $432  :  (?)=«84.29,  B     « 

820  :  192  =  $432  :  (?)  =  $101.15,  C  " 

820  :  324=$432  :  (?)=$170.69,  D  « 
Ex..  5.     $6+$4+$2  =  $12 

$2640  X  T%=$1320,  B's  share  ; 

$2640  X  T43  =     880,  C's       « 

$2640  X  A  =     440,  D;s      " 
Ex.  6.     $6300X4=^900,  A?s;  $6300  x  £  =  1260,  B  s; 

$6300  X  §=$1400,  C's;  $900+$1400=  $2300,  D's  • 

$6300—  ($900+$1260+$1400+$2300)  =$440 

for  E  and  F; 

$440X|=$165,  EV;  $440x| 

Ex.7,     f  ":  8  =  J:  |=4:  5;  4+5=9; 
$90X|=$40,  share  of  first  ; 
50,      "         second. 


Ex.  8  .     If  the  younger  has  8  shares, 
the  elder  will  have  9      " 


17,  sum  of  shares  ; 
(362) 


PARTNERSHIP.  183 

$5463.80  X  I8?  =  $2571.20,  the  younger  receives; 
$5463.80XT97=  2892.60,  the  eider  « 

Ex.9.     $1680  —  $840=8840,  A  and    B's  gain;    and  since 
C's  gain  is  equal  to  A  and  B's  gains  together, 
his  stock  must  be  $12000+$8000=$20000  ; 
8840  X~8o  =$336,  A's  gain; 


Ex.  10.  Since  the  portions  of  the  stock  which  they  severally 
put  in,  are  proportioned  to  their  gains  respectively, 
we  have  $2000+$2800.75+$1685.25+$1014 
=$7500,  whole  gain; 

$7500  :  $2000      =  $22500  :  (?)  =  $0000,  A's  stock  ; 
$7500  :  $2800.75  =  $22500*  (?)X  $8402.25,  B's  « 
$7500  :  $1685.25=$22500  :  (?)=$5055.75,  C's  " 
$7500  :  $1014      =$22500  :  (?)  =  $3042,  D's       « 

Ex.  11.  J  :  f  :  |  :i=i*  :  fMg  :  |{|  =  15  :  36  :  40  :  30 

15+36+40+30=121,  sum  of  proportional  terms; 

$30000XT¥T=*3719T27^  share  of  lst; 
$30000  X$h=  8925T\5T,     "      "  2d; 

$30000xT43°r=  9917TV^     "      "  8d^ 
33°,=  7438T|T,     "      «  4th. 


Ex.  12.  $71.27+$142t54=$213.81,  gain  of  A  and  B; 
$1200—  $500  =  $700,  stock  of  A  and  B; 
$700  :  .$500=1213.81  :  (?)=$152.724,  C's  gain; 
$213.81  :  $71.27=$700  :  (?)=$233.33J,  A's  stock 
$213.81  :  $142.54=1700  :  (?)=$466.66f,  B's   « 

Ex.  13.  3+5+7  =  15;  $1  8840  XT7s=  $8792,  C's  stock; 
$8792XT^=$175S.40,   A's  gain; 
$8792XT55=  2930.66f,  B's    « 
$8792XT75=  4102.931,  C's     " 
(33?,  363) 


184  PARTNERSHIP. 

Ex.  14.  I  ,  J%  ,  |§=|f  ,  £{j ,  ^  ;  and  these   fractions   are 
proportioned  to  15,  18  and  13. 
Hence,  A  and  B  receive  15  shares, 

A    «    C      "      18      « 

B    «    C      "      13      " 

~~  ^ 

A,  B  and  C  receive  4  of  46=23  shares;  and 

A  will  receive  23 — 13  =  10  shares, 

B     «       «        23—18=  5     « 

C     «       «        23—15=  8     «         .  Hence 

A  has  ^{  of  $26.45=811.50; 

B   «    ^  of  $26.45=     5.75; 

C    «    2%  of  $26.45=    9.20. 

(63O5  page  364!) 

Ex.  1.     $357X  5=$1785,  A's  product; 
371X  7=  2597,  B^s        « 
154X11=  1694,  C's         « 


$6076  :  $1785=$347.20  :  (?)  =  $102,      A's  gain; 
$6076  :  $2597  =  $3  47.20  :  (?)  =  $148.40,  B's     " 
$6076  :  $1694=$347.20  :  (?)=$  96.80,  C's    " 

Ex.  2.     5x12=60  cows  for  1  week=A^s  use  of  pasture; 
4X10=40     «      «         "     =B's    «         " 
6X  8=48     «      «        «     =C's    «         " 

148     "      "        " 
$55.50  Xi^^22-50*  A  pays 

«55.50XT4A=  15     >  B    " 
$55.50  Xi4485=  18     ,C     " 

(363,  364) 


PARTNERSHIP.  185 

Ex.  3.     $4500— $1800=$2700,  B's  gain ; 

$15000X12=1180000,  B's  capital  for  1  mo.; 
$2700-*-180000=$.015  gain  per  month  on  $1 ; 
$.015X9=1135,  gain  on  $1  for  9  mo.,  or  C's  time; 
$1800-7-.135:=$13333.33|,  value  of  C's  land; 
$13333|-f-125=$106f ,  value  of  land  per  acre,  An*. 

Ex.4.    $4200x9=137800  $1500x  6=$  9000 

$4400x7=  30800  $1000x10=  10000 


A's  product,  $68600;  B's  product,  $19000 

$68600+$19000=$87600,  sum  of  products; 
876  :  $686=1772.20  :  (?) =$604.71,  A's  gain; 
:  $190=$772.20  :  (?)=$167.49,  B's    « 


Ex.  5.     30X12X  9=3240  hours'  work  by  1st  company; 
32X1^X10=4800     "          "      «  2d        « 
28X18X11=5544    «         '«      «  3d        « 
20X15X12=3600     «          «      «  4th       « 


17184     «          "     *^  all. 

$1500XT372T48°4  -—$282.82,  wages  of  1st  company ; 
$1500XT478T°8°4=  418.99,      «  .  J<  2d        « 
$1500Xffr4s\=  483.94,      «    ^  3d        « 
$1500  xT3T6T°g°4=  314.25,      «      "  4th      « 

Ex.  6.     To  avoid  fractions,  suppose  A  had  $5X^=$30,  and 
Bhad$8x6=$48.     Then 
$30X4=1120  $48  x  4= $192 

$15x8-f-  120  $16X8=  128 


$240,  A's  prod.  ;  $320,  B's  prod.  ; 

$240+$320=$560,  sum  of  products; 


=  2285f,  B's. 
(364,  365^ 


186  PARTNERSHIP. 

Ex.  7.  $15800X4=$  63200 
14600X2=  29200 
13100X4=  52400 
14100x8=  112800  $257600,  B's  prod.; 

$25000X6=$150000 
23000X4=  92000 
21500x1=  21500 
22300X7=  156100  419600,  C's  prod. ; 

$30000XT=$210000 
31800X3=  95400 
26800X8=  214400  519800,  D's  prod. ; 

$1197000,  sum  of  products; 
$15000xT\Wo=$3228.07,  B  receives; 
k  $15000  XT4TWo=  5258.15,  C        « 

$15000 xT6TW%=  6513.78,  D        « 
Ex.  8      C  put  in  1 — (i+|j=t\j  f°r  1  Jear;  hence 
£X|=  J?  A's  product; 


But  |?  i  ^nd  TV=|i  4%  and  4%  5  and  these  frac- 
tions  are  proportional  to  15,  8  and  4 ;  hence 
15+8+4=27,  sum  of  proportional  terms; 
$5400X;Mf =$3000,  A's  share  of  the  gain; 
$5400X28T=  1600,  B's     "      ""       « 
$5400X347=     800,  C's      «      «  «      " 
$5400 X  i=  2700,  stock  A  put  in; 
$5400  X  |=  2160,     «    B    «     « 
$5400  X71iy=     ^40,     "     G    «    « 
$3000+$2700=$5700,  A's  share  of  entire  stock; 
$1600+$2160=  3760,  B's     «     «      "        " 
$800+$540  =  1340,  C's     «     '«      «        « 
(365) 


PARTNERSHIP.  187 

Ex.  9.     $456-=-10      =$45.60,  A's  monthly  profit; 
$343.20-r-8  =  42.90,  B's      "  " 

$750-f-12      =  62.50,  C  s       «  " 

Now,  their  respective  amounts  of  capital  must  be  pro- 
portional to  their  monthly  profits  ;  hence 
$45.60+$42.90+$62.50=$151  ; 
$151  :  $45.60  =$14345  :  (?)=$4332,    A's  capital; 
$151  :  $42.90=814345  :  (?)  =  $4075.50,  B's     " 
$151  :  $62.50=$14345  :  (?)=$5937.50,  C's     « 

Ex.  10.  C  will  have  1—  (T^+|)—  ^  of  the  profits; 
y1^  -5-4=^5,  B  may  claim  for  1  mo.; 
i-f-8=A,0     «       «      «      « 
|-j-6  =TJB,  D    "        "       "      "     ;  hence 
j\  :  ^=$600.0  :  (?)  =  $2250,  B  put  in; 
(?)  =  $5625,  C    «     « 


Ex.  11.  $2400—  $1920  =$480,  A's  gain; 
$2080—  $1280=  800,  C's    « 
Now,  since  the  gain  varies  as  the  product  of  the  cap- 
ital and  time,  we  have  the  compound  proportion 

1920  (  1280 

:        )  =$480  :  $800 

6  /     (?) 

(?)=iff§g^f  00^15  mo.,  O's  time. 
Again,  since  A  gained  $480  in  6  mo.,  he  would  have 
gained  $480x2=$960  in  12  mo.,  B;s  time;  and 
his  stock  and  gain  together  would  have  been  $1920 
+$960=$2880;  hence 
$2880  :  $1920=$4800  :  (?)=$3200;  B's  stock. 

(365) 


188 


ALLIGATION. 


Ex.1. 


ALLIGATION. 
(633,  page  366.) 


$  .60X4=$2.40 

.70X3=  2.10 

1.10X1=  1.10 

1.20X2=  2.40 


Ex.2. 


SO  X14=$  0.00 
.75X12=  9.00 
.90X24=  21.60 

1.10X16=  17.60 


10  )  $8.00 


66  )  $48.20 


Ans.    $.80 

Ex.  3.     3  Ib.  6  oz.=42  oz. ; 
4  "    8  "  =56  " 
3  "   9  "  =45  " 
2  "   2  "  =26  " 


Ex.4.    $1.20xl5=$18.00 

1.10X  5=     5.50 

.90  X  5=     4.50 

.70X10=     7.00 


Ans.    $.73 Jg. 

23X42=  966  carats; 
21X56=1176      « 
20X45=  900      « 
0X26=     -0      " 


169  )  3042      « 


18  carats;  Ans. 


35  )  $35.00 


worth  of  Ibu.,  $1.00  \  $1.25— $1=$.25,  Ans. 
Ex.  5.    8.05x17=$    .85 


.08X51=  4.08 
.10X68=  6.80 
.12X17=  2.04 


153    $13.77 

(366,  367) 


$13.77_,_153~$.09;average 
selling  price ; 
$.09-f-1.33|=$.0675,  Ans. 


ALLIGATION. 


189 


Ex.6. 


Ex.7. 


Ex.8. 


Ex.2. 


Ex.3. 


$2.70X42=1113.40 
2.85X48=  136.80 
8.24X65=  210.60 


155     $460.80,  cost; 
$460.80X1^=1552.96,  to  be  sold  for; 
$552.9C-:-155=$3.567if,  Ans. 

As  the  degrees  and  minutes  of  all  the  observations 
are  alike,  we  need  to  average  only  the  seconds,  thus : 
25.4"+24.5"+27.8"+26.9"+25.4"+24.7" 
+24.2"+26.3"+25.8"+26.7"=257.7"; 
257.7"-*-10=25.77".    Ans.  36°  17'  25.77". 

Since  the  third  trial  is  entitled  to  twice  the  degree 
of  reliance  to  be  placed  upon  either  of  the  others,  it 
is  equivalent  to  two  trials  giving  the  same  result ; 
hence  37  min.  54.16 sec.,  1st  trial; 

37    «     55.56    «   2d     « 

37    "     54.82    "   3d      " 

37    "     54.82    «  3d     « 


44  )  151  min.  39.36  sec. 

37  min.  54.84  sec.=9°  28'  42.6",  Ans. 

(634,  page  370.^ 


40 

8 

A 

15 

3 

A 

15 

3 

75 

15 

Ans.  3  Ib.  of  the  first 
kind,  2  Ib.  of  the  second, 
3  Ib.  of  the  third,  and  5 
Ib.  of  the  fourth. 

Ans.  8  gal.  of  water  to  3 
gal.  at  $.60, 3  gal.  at  $.90, 
and  15  gal.  at  $1.15. 


(367—370) 


190                                      ALLIGATION. 

Ex.4. 

'  , 

1 

40 



17^ 

17^ 

35 

7 

22.  * 

1 

62^ 

60 



17i 

V* 

35 

7 

1 

f 

80 





22^ 

22 

25 

50 

10 

fc 

17* 

17i 

Sx.  5. 


Ex.6. 


Ans.   7  A.  from  each  of  the  first  two  tracts,  and  10 
A.  from  the  other. 


{60 

_i_ 

15 

15 

3 

50 

^ 

I 

7 

3 

10 

2 

42 

i 

5 

5 

1 

38 

4 

5 

5 

1 

30 

_i 

15 

15 

3 

Ans.  3  pt.  of  the  first  kind,  2  pt.  of  the  second,  1  pfc. 
of  the  third,  1  pt.  of  the  fourth  and  3  pt.  of  fifth. 


Another  Solution 

1 

1 

i 

i 

7 

1 

8 

5 

5 

i 
57 

5 

5 

T5 

3 

3 

Ans.  1  pt.  of  first  kind,  8  pt.  of  second,  5  pt.  of  third, 
5  pt.  of  fourth  and  3  pt.  of  fifth. 


40 


24- 

24 

8 

Ans.  3  Ib 

atf 

24 

40 

120 

24 

160 

3 
20 

pure,  3  Ib. 

at   § 

pure,  20  Ib. 

iV 

NOTE. — In  the  above  solution,  we  multiply  the  first  ratio,  8  :  40, 
by  3,  making  24  :  120.  Then  by  adding-,  we  get  24  :  24  :  160, 
which  reduced  gives  3  :  3  :  20. 


(370) 


ALLIGATION. 


191 


(635,  page  371.) 


Ex.1 


58  - 


25 

j& 

12 

4 

4 

NOTE.  —  Divide  the 

50 

1 

4 

4 

terms  in  the  3d  en!  - 

62 

8 

8 

umn  by  3,  and  to 

70 

A 

4 

33 

11 

11 

the  result  add  the 
terms  in  the  4th. 

Ans.  4  -gal.  of  the  first  two  kinds,  8  gal.   of  the 
third,  and  11  gal.  of  the  fourth. 


Ex.2. 


Ans.  5,  5,  5,  and  2. 


NOTE. --Multiply  the 
terms  in  the  third  col- 
umn by  5,  and  add  the 
result  to  the  terms  in  the 
fourth  column. 


Ex.3. 


For  the  first  mixture. 


C   30 

70-^    50 

(.100 


3TJ 


Ans.  1,  1,  and  2. 


For  the  second  mixture. 
3 


3 

1 

3 

12 

12 

4 

2 

8 

12 

4 

Ans. 

1,  4  and  4. 


70  \    50 
(100 

NOTE. — Multiply  the  terms  in  the  fourth  column  by  4,  and  to 
the  result  add  the  terms  in  the  third  column,  and  then  divide  by  3. 

(636,  page  372.) 

Ans.  10  at 
$16, 10  at  $18 
and  60  at  $24. 

Ex.  2.     $15-r-12r=$l|,  price   per  yard  of  the   ingredient 
whose  quantity  is  limited. 


Ex.  1.      (  16 

I 

1 

1  f  10 

22  J  18 
22J20 

1 

, 

1 

1 

1 

10 
10 

[24 

I 

\ 

| 

3 

2 

1 

6 

60 

1 

11 

12 

4 

16 

Ans. 

16yd. 

9 

9 

9 
9 

3 

3 

12 
12 

at$|, 

and  12 

yd.$: 

ff- 

(371,  372N 

192 


ALLIGATION. 


i:x.  3.     9  gal.  2  qt.  1  pt.=77  pt. 


"{„$ 

SI* 

11  1 
77  1  Ans.  11  pt.  =  l  gal.  1  qt.  1  pi 

Ex.4.   ("30 
45-}  55 

A 

A 

T5 

2 
3 

5 

10 
15 

15 
15 

60 
60 

(70 

sV 

3 

3 

12 

Ans.  60  Ib. 

(637,  page  373.) 

Ex.  1.     $2.20X7=115.40 
$2.00X7-*-  14.00 


14  )  §29.40 


$2.10,  average  price  of  the  14  yd. 


(160 

180  i  175 

(210 


A 


14 

28 
14 


Ans.  14  yd, 
at  $1.60,  and 
28  yd.  $1.75. 


Ex.  2.     $1.14X60=  $68.40 
$1.26X30=     37.80 


90  )  $106.20 


$1.18,  average  price  of  the  90  gaL 


157  { 


118 
175 


18 
39 


90  I 
195  I  Ans.  195  gal. 


Ex.  3.     $2.00X40=  $80 
$  .50X70=     35 


110  )  $115 


u. 


22 


$  IT^,  average  price  of  the  110  bu. 
4  I    2  |    6| 


66     33     110 


Ans.  Q'i  bu. 


NOTH. — Multiply  the  terms  in  the  fourth  column  by  *££  =  * 
(372,  373) 


INVOLUTION. 

(638,  page  374  ) 


193 


Ex.  1.      f   8 

T2 

1 

1 

40     Ans.  40  Ib.  at  8 

16 
20^  24 

I 

j 

^ 

1 
3     1 

1 

4 

^     cts.,  40  Ib.  at  16 

4 

cts.,  and  160  Ib. 

I 

6 

240     at  24  cts. 

Ex.  2.     $154-§-154=$l,  average  price. 

' 

1 

31 

— 

1 

« 

!-< 

1* 

3 

1 

14 

i 

2 

2 

5      3 

7 

42 

98 

Ans.  14  calves,  42 

— 



sheep,  and  98  lambs. 

11 

154 

Ex.  3.     $165-i-55=  $3,  average  weekly  wages. 

f5 

^ 

2       5 

1       6 

30 

1 

^ 

1      1 

5 

4 

1 

3< 

j 

— 

4 

4 

20 

22 

Ans.   30   men,  5 

» 

11 

55 

women,  and  20  boys. 

INVOLUTION. 

(644,  page  376.) 
Ex.  3.  Ans.  2102500 
Ex.  13.  Ans.  2023.37890625 
Ex.  14.  Ans.  3838.28125 
Ex.  16.  Ans.  1871688T7T 


17 


(374—376) 


194  EVOLUTION. 

EVOLUTION. 

SQUARE   ROOT. 

(657,  page  381.) 
Ex.  3.     AM.  7502;  24.4315+. 
Ex.  9.     99225—63504=35721  ;  1/35721=189,  AM. 

Ex.10.  1/.126736=.356;  1/.045369  =  .213; 
.356—  .213  =  .143,  Ans. 

1/169  _    1/7056 

Ex.  11.  1/jf  |=       -=11  ;  I/Iff  |=        -=11  ; 
1/196  1/9216 

1  3\/  84  _  1  3 

- 


Ex.  12.     1/813X6252X24=81X625X22=202500,  AM.. 


CONTRACTED    METHOD. 


(658,  page  382.) 

Ex.  1.       |5.6568542+,  Ans.     Ex.  2.     |3.4641016+,  Jns. 
32.000000  12.000000 

25  9 


106     7.00  64     300 

636  256 


1125     6400         686     4440 
5625  4116 


11306     77500        6924     28400 
67836  27696 


(381-383) 


11312 

SQUARE 

9664 
9050 

ROOT. 

6928 

195 

704 
693 

1131 

614 
566 

693 

11 

7 

113 

48 
45 

7 

Ex.4. 
.5= 

144 

4 
4 

|.745355+,  Ans. 

11 
Ex.3. 

107 

3 
2 

J57.3322+,  Am. 

3286.9835 
25 

.555555+ 
49 

786 
749 

655 
576 

1143 

3798 
3429 

1485 

7955 

7425 

1146 

369 
344 

1490 

530 
447 

115 

25 
23 

149 

83 
75 

12 

Ex.5. 

64= 

45 

2 
2 

|2.563479+,  Ans. 

15 

Ex.6. 
1.06«  = 

21 

8 
8 

|1.156817+,  An* 

6.571428+ 
4 

1.338226:+: 
1 

257 
225 

33 
21 

(383) 


196 

506 

EVOLUTION. 

3214            225 
3036 

1282 
1125 

5123 

17828 
15369 

2306 

15726 
13836 

5126 

2459 
2050 

2312 

1890 
1850 

513 

409 
359 

231 

40 
23 

51 
Ex.7. 

201 

50 
46 

1.01258=1.0380+ 

|1.0188+,  Am. 

23 
Ex.8. 

201 

17 

16 

|1.011620  ±,An*. 

1.023375 
1 

1.0380+ 
1 

0233 
201 

380 
201 

2021 

3275 
2021 

202 

179 
162 

2022 

1254 
1213 

20 
Ex.9. 

17 
16 

202 

41 
40 

20 

1 

CUBE  BOOT. 
(661  ,  page  388.) 
-y  134217728=512;  1^512=8, 

Ans. 

Ex.10.  393042 =1544804416;  ifl 544804416= 1156,  Am. 
(383-388) 


CUBE    ROOT. 


197 


EX.  11. 


Ex.  12. 


1^50=3.6840+ 
^31=3.1413+ 

6.8253+,  sum  of  cube  roots; 
1^50-r31=^/8T=4.3267+,  cube  root  of  sum; 

2.4986+,  AIM. 


Ex.1. 


CONTRACTED   METHOD. 


(622,  page  390.) 

[2.8844992:+,  Am. 
24.000000 


68    544 

1200 
1744 

16000 
13952 

848   6784 

235200 
241984 

2048000 
1935872 

864   346 

248832 
249178 

112128 
99671 

0       4 

24952 
24956 

12457 

9982 

2496 

2475 
2246 

250 

229 
225 

25 

4 
5 

(388—390) 


198 
Ex.2. 


Ex.3. 


EVOLUTION. 

Ans,  |22.894801334±, 


62      124 

1200 
1324 

12000.812161 

8 

4000 
2648 

668    5344 

145200 
150544 

1352812 
1204352 

6849   61641 

15595200 
15656841 

148460161 
140911569 

6867    2747 

15718563 
15721310 

7548592 

6288524 

69      55 

1572406 
1572461 

1260068 
1257969 

157252 

2099 
1573 

1573 

526 

472 

157 

54 
47 

16 

7 
6 

|.555554730±r,  Am. 


.171467000 
125 


155   775 

7500 

8275 

46467 
41375 

1655  8275 

907500 
915775 

5092000 

4578875 

(390) 


CUBE   ROOT. 


Ex.4. 


Ex.5. 


1665   833 

924075 
924908 

513125 
462454 

17      9 

33     99 

92574 
92583 

50671 
46292 

9259 

4379 
3704 

926 

675 

648 

93 

27 

27 

9 

300 
399 

0 
|1.34442zb,  Am. 

2.429990 
1 

1429 
1197 

394   1576 

50700 
52276 

232990 
209104 

402   161 

53868 
54029 

23886 
21612 

4      2 

66    396 

5419 
5421 

2274 
2168 

542 

1200 
1596 

106 

108 

|2.6888±,  An*. 

19.440 

8 

11440 
9576 

(390) 


200 


EVOLUTION. 


Ex.6. 


Ex.7. 


78     60 

2028 
2088 

1864 
1670 

1      1 

274   1096 

215 
216 

194 
173 

22 

21 

18 

2 

1= 

24300 
25396 

3 
|.941035±,  Ans. 

.833333 
729 

104333 
101584 

282    28 

26508 
26536 

2749 
2654 

242     41 

2656 

95 

80 

27 

19200 
34  19684 

15 
14 

|.829826686±,  An*. 

.571428888 
512 

59428 
39368 

2017200 
2469   22221  2039421 

20060888 
18354789 

2061723 
2487    1990  2063713 

1706099 
1650970 

206570 
25       5  206575 

55129 
41315 

(390; 


CUBE  ROOT. 


201 


Ex.8. 


Ex.9. 


20658 

13814 
12395 

2066 

1419 
1240 

207 

179 
166 

21 

13 
13 

|1.Q57023±:,  Am. 

1.08674325*=     1.181011+: 
1 


805   1525 

30000 
31525 

181011 
157625 

315   221 

308   2464 

33075 
33296 

23386 
23307 

3352 

79 

67 

34 

1.055  = 

30000 
32464 

12 
10 

|1.084715±:,  An* 

1.276282 
1 

276282 
259712 

824    130 

34992 
35122 

16570 
14049 

3      2 

3525 
3527 

2521 
2469 

353 

52 
35 

35 

17 

18 

(391) 


202  EVOLUTION. 

ROOTS    OF    ANY    DEGREE. 

(664,  page  391.) 
Ex.  1.     6=3X2;  ^6321363049  =  1849;  1/1849=43, 

Ex.2.    4=2X2;  1/5636405776=75076; 
1/75076=274,  An*. 

Ex.  3.     8=2X2X2;  1/1099511627776=1048576  ; 
1/1048576=1024;  1/1024=32,  '  Ans. 

Ex.4.     6=3X2;  1/25632972850442049=160103007; 
•^160103007=543,  Ans. 

Ex.  5.    9=3X3;  ^1.577635=1.164132+  ; 
if  1.164132=1.051963+,  Ans. 

Ex.6.     12=2X2X3;  1^16.3939=2.5404+; 

1/275404=1.5938+;  1/L5938=1.2624+,  Ans. 
Ex.7.     18=2X3X3;  1/104.9617=10.24508+; 

^10.24508=2.171893+  ; 
^2171893=1.2950+,  Am. 

(665,  page  393.) 

Ex  2.     Vl20=3.31+;  ^120=2.22+; 

3.31+2.22  =  5.53;  5.53-4-2=2.76,  assumed  root; 

2.764=58.06+  ;  120-4-58.06=2.0669+; 
2.76x4+2.0669=13.1069; 
13.1069-4-5=2.6214;  1st  approximation. 

2.62144=47.2203+  ;   120-^47.2203  =  2.54128+  ; 
2.6214  X  4  +  2.54128  =  13.02688  ; 
13.02688-^-5=2.60537,  2d  approximation. 


(391-393) 


ROOTS    OF   ANY    DEGKKE.  203 

2.604378+ ;  2.60537X4+2.604378=13.025858; 
13.025858-5-5=2.605171,  3d  approximation,  which 
is  correct  to  the  last  decimal  place. 

Ex.  3.     Vl.95u7b=1.11838+ ;  ^1.95678=1.08292+  ; 
1.11838+1.08292=2.20130 ; 
2.20130-5-2=1.10065,  assumed  root; 

L100656 =1.77785129+ ;  1.95678-5-1.77785129  = 
1.10064324+ ;  1.10065x6+1.10064324= 
7.70454324;  7.70454324-*-7=1.10064903,  1st  ap- 
proximation, correct  to  the  last  decimal  place. 

Ex.4.     10=2X5;  1/743044=862. 

Take  4  =the  assumed  5th  root  of  862;  then 
44=256;  862-5-256=3.36;  4x4+3.36=19.36; 
19.36-j-5=3.872,  1st  approximation. 

3.8724=224.771579^;  862-5-224.77 1579  =~~"~ 
3.83500443+ ;  3.872x4+3.83500443=  y 

19.32300443;  19.32300443-5-5=3.8646008,  2d 
approximation,  Ans. 

Ex.5.     15=3X5;  ^15 =2.466212+. 

Take  1.2=the  assumed  5th  root  of  2.466212 ;  then 
1.24=2.0736;  2.466212-5-2.0736=1.189339+; 
1.2x4+1.189339  =  5.989339 ; 
5.989339-5-5  =  1.197868,  1st  approximation. 

1.1978684=2.058898+ ;  2.466212-5-2.058898  = 
1.197831+ ;  1.197868  x4+1.197831=5.989303 ; 
5.989303^-5=1.197861,  2d  approximation,  correct 
to  the  last  place  ;  hence  Ans.,  1. 197861 +. 

Ex.  6.  Since  25  =  5x5,  we  might  extract  the  5th  root  of 
fye  5th  root,  for  the  25th  root.  A  more  convenient 
method  is  as  follows : 

(393,) 


204  EVOLUTION.        - 

1/100=10; 

VlOO  =1/10=3.1622+; 

VlOO  =1/3.1622=1.7782+; 

VTo6=f/1.7782=1.2115+. 

Now,  as  the  25th  root  must  be  less  than  the  24th 

root,  take  1.2= the  assumed  root. 

1.224  =79.49684+  ;   100-5-79.49684=1.25792+  ; 

1.2x24+1.25792=30.05792 ; 

30.05792-^-25=1.2023168,  1st  approximation. 

1.202316824=83.2677184+ ;  100--83.2677184= 
1.2009492+ ;  1.2023168x24+1.2009492= 
30.0565524 ;  30.0565524-5-25=1.202262+, 
Ans.j  correct  to  5  decimal  places. 

Ex.  7.     VT-1.5+;  V~5  =1.3+;  1.5+1.3=2.8; 
2.8-r-2=1.4,  assumed  root 

1.44=3.8416;    5-5-3.8416=1.3016+;    1.4x4+ 

1.3016=6.9016; 

6.9016-^-5= 1.38032, 1st  approximation. 

1.380324=3.63011+;  5-^-3.63011=1.37721+  ; 

1.38032x4+1.37721  =  6.89849; 
6.89849-^-5  =  1.37970+,  Ans. 


APPLICATIONS  OF  SQUARE  AND  CUBE  ROOT. 

(689,  page  396.) 

Ex.  1.     2562 =65536,  square  of  hypotenuse ; 
752=  5625,       "      «   altitude; 

59911,  square  of  base; 
(393-396) 


APltlCATIONS   OF    ROOTS.  205 

1/599H= 244.76+,  base; 
244.76  ft.— 22  ft.=222.76  ft.,  Ans. 

Ex.  2.     l/84*+502  =97.75+  lea.=337.23+  mi.,  Ans. 

Ex.  3.     1/50*— 30*  =40  ft.  from  foot  ladder  to  one  side; 
1/502— 40^=30  "      "      "        a     the  other  " 
40  ft.+30  ft.=70  ft.,  Ans. 

Ex.4.  V  (3  ~+62  = diagonal  of  one  side;  and  since  this  di- 
agonal and  the  adjacent  edge  form  a  right-angled 
triangle  whose  hypotenuse  is  the  required  diagonal, 
we  have i/6*+62+63  =1/108=10.39+  ft.,  Ans. 

(69O,  page  397.) 
Ex.  1.     208X13=2704;  1/2704=52  rods,  Ans. 

Ex.  2.  (216+24)  x  2=480  rd.  of  fence  to  inclose  the  farm  in 
rectangular  form.     $312-i-480  =  $.65,  price  per  rd. ; 
216X24=5184;  1/5184=72  rods,  in  one  side  of 
square  farm  of  same  area ;  hence  $.65  X  72  X4= 
$187.20,  cost  offence  for  square  farm; 
$312— $187.20  =  $124.80,  Ans. 

Ex.  3.     1/588><I2=84,  Ans. 

Ex.  4.     If  A  receives  |  of  the  gain,  B  receives  the  other  f ; 
consequently  B's  product  must  be  2  times  A;s  prod- 
uct.    Hence,  540x480x2=518400,  B's  product, 
which  is  the  square  of  B's  capital.     Therefore, 
1/518400--$720,  Ans. 

(691 5  page  398.) 
Ex.1.     4X3  =  12;  432--12=36;  1/36=6; 

4X6=24  ft.,  length;  3x6=18  ft.,  breadth,  Ant. 

Ex.  2.     2X3=6;  23-5-6=3.833333+; 
1/3.833333=1.95789+; 
2Xl.95789=3.91578+;|    , 
3X1.95789=5.87367+ ;  j  ^ 
(396-398) 


206  EVOLUTION.      P 

Ex.  3.     283A.  2E.  27P.=45387P.;  1x3=3  ;  45387-4-3= 
15129;  1/15129=123; 
1X123  =  123  rods,  width;   \  A 
3X123=369  rods,  length ;  j  ^ 

(692,  page  399.) 

Ex.  2.     1  :  5=752  :  (?)  =28125,  the  square  of  the  cost; 
hence   1/2^25 =$167. 70+,  Ans. 

Ex.  3.     80  :  1200 =6^__(?)  =  540,  square  of  the  diameter 
required  ;  1/540  =  23.23+ft.,  Ans. 

Ex.  4.     5h.= 18000  sec.;  55  min.  6  sec. =3306  sec.     *Since 
the  greater  the  diameter  of  the  pipe,  the  less  the 
time,  the  times  will  be  inversely  as  the  squares  of 
the  diameters.     Hence 
3306  :  18000=1.53  :  (?)=12.25+in. 
1/12.25=3.5  in.  nearly,  Ans. 

(693,  page  399.) 

Ex.  1.     24X18X4=1728  cu.  ft.,  contents; 

^1728 =12  ft.,  Ans. 
Ex.  2.     2150.4  cu.  in.Xl50=322560  cu.  in.  in  the  bin; 

1^322560=68.6  in.=5  ft.  8.6  in.,  Am. 

Ex.  3.     231  cu  in.x31.5x200  =  1455300cu.in.iTitbecistcrn; 
1^1455300=113.3  in.=9  ft.  5.3  in.,  Ans. 

Ex.  4.     ^79507=43  ft.,  length  of  one  side ; 
43-x6=11094.,  Ans. 

(694,  page  400.) 

Ex.1.     2X3X4=24;  3000--24=125;  ^126=5; 
2X5=10;  3X5  =  15;  and  4x5=20. 

Ex.2.     2X5X7  =  70;  4480-^70=64;  f 64=4  ; 
2X4=8;  5X4=20;  7x4=28. 

(398-400) 


PROMISCUOUS   EXAMPLES.  207 


Ex.  3.     2  x2jx3  =  15  ;  100-r-15=6.66666+  ; 

1^6.666666=1.882072+ ; 

2  Xl.882072=3.76414+O 
2^X1.882072=4.70518+;  [  Ans. 

3  Xl.882072=5.64622— ;j 

Ex.  4.     2150.4  cu.  in.  X450=967680  cu.  in.,  contents; 
lXlX3=3,  product  of  proportional  terms  ; 
967680-^3=322560; 

^322560=68.6  in.=5  ft.  8.6  in.,  width  and  depth; 
68.6  in.x3=205.8  in.=17  ft.  1.8  in.,  length. 


PROMISCUOUS   EXAMPLES. 

(Page  401.) 

Ex.  1.     10 A.  2R.  20P.=1700  sq.  rd. ; 

3X4=12,  product  of  proportional  terms; 

1700--12=141.66+  ;   T/14L66=11.9+ ; 

3X11.9=35.7  rd.,  breadth; 

4X11.9=47.6  rd.,  lengtK, 

35.72=1274.49 

47.62 =2265.76 

3540.25,  sum  of  squares  of  the  two  sides ; 

1/3540.25=59.5  rd.,  length  of  the  diagonal ; 

35.7  rd.+47.6  rd.=83.3  rd.,  distance  by  the  walk ; 

83.3  rd.— 59.5  rd.=23.8  rd.,  excess  ; 

3mi.x320=960rd.;  960-f-60=16rd.,rateper  min.; 

23.8^-16=1.4875  min.  =  l  min.  29.3  sec.,  Am. 
Ex.  2.     40  A.=6400  sq.  rd. ;  1/6400= 80  rd.,  Ans. 
Ex.  3.     The  dimensions  of  the  lots  are  as  100  :  144,  or  as  25 

:  36.     Now,  if  the  lots  were  25  ft.  deep  and  36  ft. 

front,  the  area  of  each  would  be  25x36=900  sq. 

ft. ',  and  a  square  lot  of  the  same  area  must  be  1/900 

(400,  401) 


208  EVOLUTION". 

=30  ft.  on  a  side.     That  is,  the  streets  must  be  30  ft, 
—  25  ft.  =  5  ft.  farther  apart.     Hence 
5  ft.  :  25  ft.  =20  ft.  :  (?)=100  ft.,  Am. 

Ex.4      40  A.  =6400  sq.  rd; 

l/6400=80'rd.,  one  side  of  the  square; 
$1.40X80X4=1448,  cost  of  fencing  the  jquare; 
6400X1  =  1600  ;  1/T6(H)=40,  breadth  of  rectangle; 
40X4=160,  length  of  " 

(160+40)  X  2  =400  rd.,  distance  round         " 
$1.40X400=1560,  cost  of  fencing  " 

$560—  $448=$ll2,  Ans. 

Ex.  5.     62  :  182=80  bbl.;  (?)=720  bbl.,  Ans. 

Ex.  6.     5^X7=38.5  sq.  ft.,  area  of  rectangle; 

1/38.5=6.20+  ft.=6  ft.  2.4+  in.,  Ans. 

Ex.  7.     The  diameter  of  the  circle  will  be  the 
diagonal  of  the  square.     Hence  42  2 
=1764,  twice  the  square  of  one  side! 
of  the  inclosed  square.  \  , 

1764-f-2=8S4,  the  square  of  one  side 
of  the  inclosed  square;  T/88i=29.7  —  in.,  Ans. 

Ex.  8.     3  A.  86  P.  =  566  P.;  17  A.  110  P,=2830  P.; 

566  :  283Q=752  :  (?)=28125,  the  square  of  the  IB- 
quired  cost;  T/28125=$167.70,  ^is. 

Ex.  9.     6  ft.X^T^6  ft-X4=3  ft.,  Ans. 
Ex.  10    123^-4s^l728-;-64=27,  Ans. 

Ex.11      8      80     16;  163 

—  =—  =-     900  X  --  =1677  Ib.  14+  oz.,  Am. 
6.5     65     13;  13  * 


Ex.  12.       |||o=-f|f  =^+=1.256+  ; 

3  in.  XL256=3.77—  in.,  Ans. 

Ex.  13.  V  Y=1^~5~r=1.71+;  4  ft.Xl.71  =  6.84  ft.,  Ans. 
(401,  402) 


ARITHMETICAL   PROGRESSION.  209 

SEEIES. 

GENERAL  PROBLEMS  IN  ARITHMETICAL  PROGRESSION, 

(7O6,  page  404.) 

Ex.  1.  (8—  1)X4=28;  28+5=33,  Am. 

Ex.  2.  (50—  1)X3  =  147;  147+2=149,  Am. 

Ex.  3.  (13—  1)+7=84;  100—84=16,  Am. 

Ex.  4.  (20-l)X=7 


(7O7,  page  405.) 

Ex.  1.  (15—  3)--(7—  1)=2,  Am. 

Ex.  2.  (51—  l)-*-(76—  1)=|,  ^ws. 

Ex.  3.  .1—  .05=.05;  8—1=7;  .05-8-7=.  00714285,  Am. 

Ex.  4.  2i--17=364,  Am. 

(7O8,  page  405.) 

Ex.  1.  75—5=70;  70-^-5=14;  14+1=15,  Am. 
Ex.  2.  20~£  =  19i;  19^6*  =3;  3+1=4,  Am. 
Ex.  3.  2.5—  .25=2.25  ;  2.25-f-.125=18  ;  18+1=19,  Am. 

Ex.  4.  If  there  are  5  means,  there  will  be  5+2=7  terms. 
Hence  37  —  2=35  ;  35-^-6=  5|,  common  difference. 
Am.  2,  7|,  13f  ,  19^,  25|,  31J,  37. 

(TOO,  page  406.) 

Ex.  1  .  40—4=36  :  36-~6=6;  6+1=7,  number  of  terms; 
(40+4)  X  1^=154,  Am. 

Ex.  2      250  X  H0-  =  !  25000>  -4^- 

Ex.  3.     ll=N"o.  terms  ;  220=last  term  ;  17=com.  diff.  ; 

(11—  1)  XI  7=  170;  $220—  $170=$50,  first  term,  • 

($220+$50)XV  =11485. 

(404-406) 


210  SERIES. 

GENERAL    PROBLEMS    IN  GEOMETRICAL   PROGRESSION. 

(711  ,  page  407.) 

Ex.1.     6X46  =6144,  Ans.      Ex.2.     192-*-2  <*=  3,  Ans. 

122 

Ex.  3.     6X-—  =—  =  -  ,  Ans. 
37     36     729 

1     25     5*      1 

Ex.4.     25X—  =—=—=—,  An*., 
54     54     54     25 

(712,  page  408.) 
Ex.1.     512n-2=256;    V256==4, 


Ex.  3.  7--.0112=:625  ;  V625=5,  Ans. 
Ex  4.  5000—8=625;  Vg25=5s  ratio; 
8,  40,  200,  1000,  5000. 


(713,  page  408.) 


Ex  1.     1458-^2=729 


Ex.  2.     100-4-1=1000 


3 

729 

3 

243 

3 

81 

3 

27- 

3 

9 

3 

3 

1    Ans  7. 

10 
10 
10 


1000 


100 


10 


(407  408) 


GEOMETRICAL    PROGRESSION. 


211 


Ex.  3 


Ex.  4.     196608-4-6=32768 


128 
64 
32 
16 


8 

32768 

8 

4096 

8 

512 

8 

64 

8 

8 

1 

Ans.  6. 

i.8. 


(714,  page  409.) 

Ex.  1.     384X2=768;  768—3=765;  765-f-l=765,  Ans. 

Ex.2.     1080X6=6480;  6480— 5=6475; 
6475^-5=1295,  Ans. 

Ex.3.    4fX3=14|;  141-^=14^1; 

Uj{j|-f-2=7tffr,  Ans. 
Ex.  4.     The  least  extreme  is  0,  and  the  ratio  is  8-j-4=2 ; 

hence,  8x2=16,  Ans. 

(7 15,  page  410.) 

Ex.1.     34=81;  81— 1=80;  80X7=560; 

560-j-2=280,  Ans. 

Ex.  2.     375-^-5 3=3,  least  term;  54=625;  625—1=624; 
624X3=1872;  1872-f-4=468,  Ans. 

Ex.  3.     1.065  =1.338226+;  175X-338226= 

59.189550 ;  59.18955-f-.06=986.49+,  Ans. 

(409,  410) 


212 

(716,  page  410.) 
Ex.  1.     800—2=798;  800—686=114;  798--114=7;^n*. 

Ex.2.     127f— 1=-127£;  127J— 64=63| ; 
127^-r-63j=2,  Ans. 

Ex.3.     4»— 0=4^;  4^—3=11; 


COMPOUND   INTEREST   BY  GEOMETRICAL   PROGRESSION. 

(718,  page  411.) 

Ex.  1.     $3  50= first  term,  5= number  of  terms,  and 
1.06= ratio ;  the  last  term  is  required. 
$350  XL064 =$441.86+,  Ans. 

Ex.  2.     To  find  the  compound  interest  of  $1,  we  have  $1= 
first  term,  1.07=ratio,  3=No.  terms; 
$lXl.072=$l'.1449,  amount; 
$1.1449— $1  =  1.1449,  int.  of  $1; 
$150-r-.1449=$1035.196+,  Ans. 

Ex.  3.  We  have  given  $1000=last  term,  1.06=ratio,  4= 
No.  terms,  to  find  the  first  term.  Reversing  the  rule, 
$1000-*-1.06»=$839.62— ,  Ans. 

Ex,  4.     $40=first  term;  $53. 24= last  term;   1.10=ratio. 
We  find  the  number  of  terms  by  Prob.  Ill,  thus  : 
53.24--40=1.331;   1.331-r-1.10  =  1.21 ;  1.21-j- 
1.10=1.10;  1.10-5-1.10=1. 
Hence  3-f-l=4=No.  terms ;  and  Ans.,  3  years. 

Ex.  5.     Let  the  principal  be  $1 ;  then  1= first  term  ; 

2= last  term ;  9=No.  terms.     We  find  the  ratio  by 
Prob.  II,  thus  :  2--l=2 ;    V~2"==1.0905+ ; 
1.0905— 1=.0905=9.05+  %,  Ans. 

(410-412) 


ANNUITIES.  213 

Ex.  6.     $322.51=  last  term,  1.05=  ratio,  25=  No.  terms. 
We  find  the  first  term  by  Prob.  I,  thus :  1.052<= 
8.225100+ ;  $322.51-f-3.2251=$100,  Ans. 


ANNUITIES   AT   SIMPLE   INTEREST 

(726,  page  414.) 

Ex.3.     $150=the  last  term;  $150X.015=$2.25=common 
difference;  5^X4=22=No.  terms.      We  find  the 
sum  of  the  series  by  TOG  and  7O9,  thus  : 
$150+($2.25X21)=$197.25,  first  term; 

$197.25+1150 

X 22 =$3819.75,  Ans. 

2 

Ex.  4.  $500= last  term,  $3450= sum  of  series,  $500  X -06= 
$30  =  common  difference;  we  are  required  to  find 
the  number  of  terms.  Now  3450-^500=6.9;  that 
is,  if  the  pension  did  not  draw  interest,  the  time  re- 
quired for  it  to  amount  to  the  given  sum  would  be 
less  than  7  yr. ;  by  trial  we  find  the  time  to  be  6  yr. 

Ex.  5.  $6000=last  term;  $59760=the  sum  of  the  series; 
8= No  of  terms.  We  find  the  common  difference, 
thus :  according  to  7O9.  2  times  the  sum  of  the 
series = the  sum  of  the  extremes  multiplied  by  the 
number  of  terms.  Therefore 

$59760X2 

=$14940,  the  sum  of  the  extremes; 

8 

$14940— $6000=88940,  first  term.     Hence  by 
7O7,  $8940— $6000 =$2940;  $2940-^-7 =$420, 
common  difference.     Then 
$420-^46000 =.07 =7  %,  Am. 
(412-414) 


214  SERIES. 

PROMISCUOUS   EXAMPLES   IN   SERIES. 
(729,  page  415.) 

Ex.  1.     Its  present  amount  will  be  the  sum  of  the  geometric- 
al series  in  which  $200=  first  term;  1.06=  ratio, 
and  20=  No  terms.     Hence,  by  715, 
200X(1.062°— 1)  200X2.207135+ 

= =$7357.11+. 

1.06—1  .06 

Ex.  2.  We  have  given  $16459.35=sum  of  the  series,  25= 
No.  of  terms,  1.06= ratio,  to  find  the  first  term. 
According  to  715,  the  sum  of  the  series  is  equal  to 

1.0635— 1- 

the  first  term  multiplied  by  the  fraction 

1.06—1 ; 

consequently  the  first  term  will  be  found  by  dividing 
the  sum  of  the  series  by  the  same  fraction ;  and  we 
1.06—1       $987.5613 

have  $16459.35  X = =$300. 

1.06^  5—1       3.291871 

Ex.  3.  We  first  find  the  amount  of  the  annuity  in  arrears 
for  the  7  years.  We  have  given  $500 = first  term, 
1.06= ratio,  and  7= No.  terms.  Hence  by  715. 
$500  X  (1.067— 1)  $251 .815 

= =$4196.911,  sum  of 

1.06—1  .06 

series.  We  now  find,  by  553,  what  sum  will 
amount  to  $4196.91|  in  7  years,  at  6%  compound 
interest;  thus:  $4196.91f-=-1.503630=$2791.18+.' 

Kx.  4.  We  first  find  the  value  of  the  annuity  in  arrears 
for  the  20  years,  or  its  worth  when  it  expires.  We 
have  given  $100= first  term,  20= No.  terms,  and 
1.05= ratio,  to  find  the  sum  of  the  series.  By  715. 

(415) 


EXAMINES.  215 

$100x(1.0520— 1 


-=$3306.596,  sum  of  series. 


1.05—1 

This  is  what  the  lease  is  worth  20-{-14=34  years 
hence  ;  therefore  its  present  value,  by  Ot>3,  is 
$3306.596-^-5.253348  =  $629.426+,  An*. 

21=  No.   terms,  5=firstterm,  |  =  common  diff. 
By  7O6,  5—  (£X20)=0,  last  term; 
by  TOO,  £+£  X21=52|,  Am. 

Ex.  6.     80  —  last  term,  5=common  diff.,  13=No.  terms;  to 
find  the  first  term,  and  the  sum  of  the  series. 
Reversing  the  rule  under  7OG,  we  have  (13  —  1)X 
5=60  ;  80—60=20,  first  day's  journey.     Then  by 
7OO,  ^^^X  13  =650,  whole  distance  traveled. 

1 

Ex.  7.     15-r-30=|,theratio;by711,30X—  =j 


Ex.  8.     By  714,  <^*f-a  =  2-°34-  =682>  Ans- 
Ex.  9.     We  have  given  360=sum  of  an  arithmetical  series, 
27=  first  term,  and  45=  last  term,  to  find  the  num- 
ber of  terms.     By  TOO,  twice  the  sum  of  the  series 
is  equal  to  the  sum  of  the  extremes  multiplied  by 
the  number  of  terms;  conversely,  ||^||=10;  Ans. 
Ex.  10.  By  712,  Vl5625=5,  Ans. 

Ex.  11.  $500=first  term,  10=  No.  terms,  1.06=  ratio;  to 
find  the  sum  of  the  series.  By 

$500  X  (1.06  1  °—i)     $395.424 
----  =  --  =$6590.40,  Ans. 
1.06—1  .06       ^ 

Ex.  12.  Reversing  the  rule  under  7OO,  we  have  ^oi^ia^lO^ 
the  sum  of  the  extremes.  Now,  since  6  is  the  com- 
mon difference,  and  8  the  number  of  terms,  (8  —  1) 

(415,  416) 


216  SERIES. 

X6=42  is  the  difference  of  the  extremes;  hence  by 
Prob.  33,  127,  ui^-hi^ 72,  last  term;  and  UlAf** 
=30,  first  term. 

Ex.  13.  We  have  $1196=  the  sum  of  an  arithmetical  series, 
13=No.  of  terms,  and  $12=.  the  common  difference, 
to  find  the  first  and  last  terms.  Proceeding  as  in 
the  last  example,  mffZ3=tIS4,  the  sum  of  the 
extremes ;  (13—1)  X 12=$144,  the  difference  of  the 
extremes.  *l£±5*il.4=$20,  first  payment ; 
•.LS4+ill.4=$164,  last  payment. 

Ex.  14.  $2=  first  term  of  a  geometrical  series,  $512=  last 
term,  and  4=  ratio ;  to  find  the  number  of  terms, 
and  the  sum  of  the  series. 

Firstly  713,  512--2=256;  256--4=64;  64-5-4 
=16;  16--4=4;  and  4 --4=1.     Hence  4+1=5, 
number  of  payments. 
Second,  by  7 14$  «Hi*±>=*==$682,  indebtedness. 

Ex.  15.  $4800=  first  term,  5=  No.  terms,  1£=  ratio;  to 
find  the  last  term,  and  the  sum  of  the  series. 
By  711 ,  $4800x(li)4=$24300,  share  of  eldest; 

($24300x1-1)— $4800 
By  714,  —  —=$63300,  property, 

2         . 

Ex.  16.  $2818.546=  sum  of  series,  5=  No.  terms,  1.06=  ra- 
tio ;  to  find  the  first  term.  Reversing  the  rule  under 
71t>,  as  in  Ex.  2,  we  have 

1.06—1       $169.11276 
$2818.546 X = -=$500,  An*. 

1.066— 1        .338226 

Ex.  17.  $10=  first  term,  $7290=  last  term,  3=  ratio,  to 
find  the  number  of  terms,  and  the  sum  of  the  series. 
By  713,  7290-f-10=729;  729--3=243; 
(416) 


PROMISCUOUS   EXAMPLES.  217 

243-j-3=81;  81-5-3=27;  27--3=9  ;  9-^3=3; 

3-j-3=l  ;  6+1=7,  payments. 

%By  714,  (•3^gA|JO=lLO=4l0930,  debt. 

Ex.  18.  If  we  take  the  distance  traveled  in  going  to  the 
several  stations  and  returning,  we  shall  have  an 
arithmetical  series,  of  which  10  mi.=the  first  term, 
50  mi.=the  last  term,  and  180  mi.=the  sum.  We 
are  to  find  the  number  of  terms  and  the  common 
difference.  Reversing  rule  under  *7OO,  we  have 
lgax2__(^  NO>  stations.  And  reversing  rule  under 
7OG.  ^-§Zj^-=8,  common  difference;  hence  8-v-2 
=4  mi.,  distance  between  the  stations. 

B*.  19.  We  first  find  the  amount  of  the  annuity  in  arrears 
for  12  years,  by  7  15. 

$200  X  (1-06  *  3—1) 

-  =$3373.99+,  amount. 
1.06—1 

Since  this  will  be  the  value  of  the  annuity  when  it 
expires,  we  must  find  its  present  worth,  at  6%  com- 
pound interest  for  6-f-  12=18  years.     By  *>*>*$, 
$3373.99--2.854339-f-$1182.05+,  Am. 

Ex.  20.  We  have  a  geometrical  series,  in  which  $6:=first 
term,  1.06=ratio,  and  60  —  16=44=No.  terms. 


Hence,  --  =$1198.548,  saved  by  dis- 
1.06—1 

pensing  with  tobacco  ;  and  $1198  544+$500= 
$1698.548,  Ans. 

Ex.  21.  The  value  of  a  perpetuity  when  entered  upon,  is  a 
sum  whose  annual  interest=the  perpetuity  ;  hence, 
S100-J-.05  %=$2000,  value  when  entered  upon. 

(416,  417) 
19 


218  SERIES. 

Since  this  is  the  value  30  years  hence,  we  find  its 
present  value  hy  OO#5  thus : 
$2000-r-4-321942=$462.75+,  Am. 

Ex.  22.  First;  $2000  is  the  amount  of  a  certain  sum  at  sim- 
ple interest  21—12=9  years  at  7%.  That  is,  $2000 
is  the  last  term  of  an  arithmetical  series,  .07  times 
the  first  term  is  the  common  difference,  and  9-}-l  is 
the  number  of  terms  ;  and  from  these  data  we  are  to 
find  the  first  term.  Assuming  a  series  of  which  $1 
is  the  first  term,  $.07  the  common  difference,  and  10 
the  number  of  terms,  by  78O?  we  find  the  last 
term  to  be  $1+($.07X10— 1)=*1.63.  Now,  since 
$1.63  is  the  last  term  of  an  arithmetical  series  oi 
which  $1  is  the  first  term,  $2000  must  be  the  last 
term  of  a  similar  series,  the  first  term  of  which  is 
$2000-5-1.63=$1226.993+,  the  sum  left  at  7% 
simple  interest. 

Second;  $2000  is  the  sum  of  a  geometrical  series, 
1.03  is  the  ratio,  and  (21— 12x2) +1=19  is  the 
number  of  terms;  and  we  are  required  to  find  the 
first  term.  By  7 1 1 ,  we  have  §2000-f-1.03 1 »- 1  = 
$2000-*-1.702433=$1174.789+,  the  sum  left  at 
6%  compound  interest  payable  semi-annually. 

Ex.  23.  The  prices  of  the  several  pieces  form  an  arithmetical 
series,  of  which  $136=sum,  $4=the  com.  diff.,  $31 
=the  last  term,  and  8=the  No.  terms.  By  7O6. 
31— (8^1X4)=3,  first  term,  or  the  value  of  the 
the  first  piece;  and  as  the  price  of  this  piece  was  $1. 
per  yard,  $3-r-$l=3  yards  in  the  first  piece.  Now, 
the  number  of  yards  in  the  several  pieces  form  a 
series,  the  first  term  of  which  is  3,  the  common  dif- 
ference is  2,  and  the  number  of  terms  is  8.  Hence, 

(417) 


MISCELLANEOUS    EXAMPLES.  219 

g_f_2  (8— 1)=17 yd.  in  longest  piece; 
U£t*x8=80   "      whole  quantity. 

We  now  have  3  yd.,  5  yd.,  7  yd.,  9  yd.,  11  yd., 
13  yd.,  15  yd.,  17  yd.,  lengths; 

$3,  $7,  $11,  $15,  $19,  $23,  $27,  $31,  prices; 

$1,  $1|,  $14,  $1|,  $1T8T,  $1{|,  $lf,  $1}4,  prices 
per  yard. 

Ex.  24.  600=sum  of  series,  2— ratio,  and  320=greatest  ex- 
treme. 

First,  to  find  the  No.  of  bushels  of  the  first  kind. 
According  to  the  rule  under  T14.  the  sum  of  the* 
series  multiplied  by  the  ratio  less  1,  is  equal  to  the 
difference  between  the  last  term  multiplied  by  the 
ratio  and  the  first  term;  hence,  600 X (2 — 1)=600, 
difference  between  the  last  term  multiplied  by  the 
ratio,  and  the  first  term.  Then,  320x2=640  ;  and 
640—600=40  bu.  of  the  first  kind. 

Next,  to  find  the  number  of  terms. 
320-^40=8  ;  8-5-2=4 ;  4-j-2=2  ;   2-j-2=l ; 
hence,  3+1=4  kinds.     Ans. 


MISCELLANEOUS  EXAMPLES. 

(Page  i!8.) 

Ex.  1.  18X2—36;  36x36=1296  sq.  ft.  in  both  sides  of 
the  roof.  But  1296  sq.  ft.=12.96  squares  of  100 
ft.  each;  and  since  1000  shingles  make  1  square, 
(282,  Note  3),  there  will  be  12.96  M.  shingles. 

Ex.  2.     70  mi.Xf  X 4X1=3.80952+  mi. 
1.375  mi.  X-73  =1.00375  mi. 


2.80577  mi.,  Ans. 
(417,  418) 


220 


MISCELLANEOUS    EXAMPLES. 


Ex.  3. 


Ex.4. 


Ex.5. 


|  =  137|,  the  whole  remainder; 
Ans. 


4612 

=-X-X-=-, 

4       9143 


s'o 


i.  3  gal.  of  water,  2 
of  cider,  4  of  wine,  and 
5  of  brandy. 

Ex.  6.     A  number  increased  by  ^,  ^  and  |  of  itself,  will  be 
^  1+i+i+i— ^-z  times  the  number;  hence  125-=- 

2T^r=60,  Ans. 

Ex.  7.  From  noon  to  midnight  is  12  hours;  and  since  the 
time  past  noon  is  f  of  the  time  to  midnight,  the 
whole  12  h.  must  be  If  times  the  time  to  midnight 
Hence  12  h.-f-lf=7  h.  12  min.,  time  to  midnight; 
12  h.— 7  h.  12  min.=4  h.  48  min.,  P.  M.,  Ans. 

Ex.  8.     $10X12=1120 
8|X  3=    26 

8=    60 

23  )  $206 


Ex.  9.     Int.  on 


$5.84-s-$.97j=6 


$8|f  ,  Ans. 

for  4  mo.  26  da.,  at  1 

Ans. 


24 

189 


217        (331 
5*=  I    2f 


234 
2| 
31 


Ex.10. 


|XV^16^.,  Ans. 

Ex.  11    $450X.05X6|=$153.75,  interest; 

$450-~1.34|=$335.40,  present  worth; 

(418) 


MISCELLANEOUS    EXAMPLES.  221 

$450—  $335.40=$114.60,  discount; 
$153.75—  $114.60=$39.15,  Am. 

Ex.  12.  $6300xf=$7200,  eldjer  brother  received  : 
$6300+17200=413500,  An*. 

Ex.  13.  .7=3;  .88=fi=l;  -727=313; 


Ex.14.  $438  X  -  -=  $547.50, 
90 

Ex.  15.   Tnt.  on  $1  for  4  mo.  3  da.=$.0239£  ; 

$1—  $.0239£=$.9760f,  proceeds  of  $1; 
$875.50--.9760f=$896.95-f-,  Ans. 

Ex.  16    $228.00^-  5=445.60,  A's  monthly  gain  ] 
266.40-7-  8=  33.30,  B?s          « 
830.00-^-12=  27.50,  C;s          « 


$106.40,  entire       « 
$2128 XTW4^*912^  A's- stock; 
2128xT3o3A=  6.66,  B'B      « 
2128Xy2o%=  550,  C?s      « 
tflx.  I1;    $.35X300=:$105,  due  Oct.  27  ; 
.40X150=     60,    «       «    31; 
.38x500=  190,    «  Nov.    7; 
.42X200=     84,    "       "    12; 
.40X250=  100,    «       «    25; 
Hence,  taking  Oct.  27  for  a  focal  date, 
105X  0=  000 
60 X  4=  240 
190=11=2090         6574^-539=12  da. ; 

84X16=1344         Oct.  27+12  da. = Nov. 
100x29=2900 


539  6574 

(418) 


222  .         MISCELLANEOUS    EXAMPLES. 

Ex.  18.  A,  B,  and  C     do    ^  of  it  in  1  day ; 
G  does  75*4  of  it,  and  A  ^ ;  hence, 
B  can  do  J2—  (^+^\)=^  in  l  da7  >  and 
408-r-5=81f  days,  Ans. 

Ex.  19.  We  here  have  an  arithmetical  series,  of  which  7= 
the  first  term,  51=  last  term,  and  4=  com.  diff. ; 
by  7O8,  5-V-7+l  =  12  da.,  time; 
and  by  7O9,  M^X  12 =348  mi.,  distance. 

Ex.  20.  $2500  X-OlSOf =$45.21— ,  bank  discount; 

$2500-4-1.0175=82457.00+,  true  present  worth; 

$2500— $2457=$43,  true  discount; 
$45.21— $43 =$2.21,  Ans. 

Ex.  21.  $5-5-1.035=$4.830+,  present  worth  of  $5; 
hence  this  is  the  better  condition  by  $4.875 — 
$4.83 =$.045. 

Ex.  22.  Since,  if  sold  at  cost,  ^  of  the  lot  should  have  sold 
for  only  ^  of  the  cost,  the  gain  on  ^  was  | — i==g 
of  the  cost;  and  ^-f-^=25  %,  Ans. 

Ex.23.    C$500  C$960 

]  :  $50=  1          :  $60 

(IF.  (  (0 

(?)  =  « g«2f«=i  yr.=7  mo.  15  da.,  Ans. 

Ex.  24.  The  remainder,  which  is  88  %  of  the  whole,  must  be 
sold  for  125  %  of  the  cost  of  the  whole ;  hence  125 
-5-88  =  1.42^;  1.42^—1=42^  %  above  cost. 

Ex,  25.   3_6oiKa6j7J>  — 182.53375,  greater; 

3 « 5-^0 6 * s -,182.46625,  less* 
Ex.  26.  7  yr.=84  mo. ;  3  yr.  lOmo.  15  da. =46£  ma.; 

(445.625         (650  _12ooo.m 
i    84  '{   46^=128.99  .  (0 

xlx||.^==$io4.15+,  Am. 
(418,  41?) 


MISCELLANEOUS    EXAMPLES.  223 

Ex.27.  £JJi*  1*4*1=:  $10125,  invoice; 

$10125Xl.26=$12757.50,  sold  for; 
$12757.50X-015=$191.36,  commission ; 
$12757.50— $191.36=$12566.14,  net  proceeds; 
$12566.14-*-.995=$12629.28+,  Ans. 

Ex.  28.  $.194X5000 =$970,  draft  on  Paris; 

5000-:-5.20=$961.53+,  remittance  from  Paris. 

Ex.  29.  $1000-r-.065=$15384.61+,  stock  required;       • 
$15384.61  Xl.05=$16153.84,  Am. 

Ex.  30.     $25000xT72%=::$14583-33ib  stock  at  8  %  ', 

$25000 X  -06=     1500.00  ,  income  of  6  %  stock; 
•14583.333  X -08=     1166.66|,      "      "  8  %      « 

$333.33^,  Ans. 

Ex.  31.  $15190--.28 =$54250,  money  of  all. 

Suppose  A  has  $1,  } 

B  will  have  |  X|  =  $f  >  \  proportionate  terms. 

o   "     "      !x|x|-$|?,) 

Multiplying  these  proportioned  terms  by  81,  we  have 
81,  72  and  80,  the  proportional  terms  of  A,  B,  and 
C  respectively. 

81-|_72+80=233,  sum  of  proportional  terms; 
854250 X^Vs^18859-4^  A's  money; 
$54250X2%=  16763.95+,  B's      " 
•54250  X  3%=  18626.61+,  C's      « 

Ex.  32.  21 — 14=7  payments,  or  deposits.     Now  the  last  de- 
posit will  not  draw  interest;  the  last  but  one  will! 
draw  compound  interest  for  2  half  years,  and  th< 
amount  will  be  $250xL032  ;  the  last  but  two  wil 
draw  compound  interest  for  4  half  years,  and  th 
amount  will  be  $250xL034 ;  and  so  on.     Hence 
we  have  $250=first  term,  7=No.  terms,  and  1.03 
(419> 


224  MISCELLANEOUS    EXAMPLES. 

=  the  ratio,  to  find  the  sum  of  the  series.     Now 
(1.032)^=1.0314;  hence  by  715~ 
(1.03 14— 1)X250 

=$2104.227+,  Ans. 

1.03—1 


Ex.  33.  | — f=3nj  cents>  profit  on  1  peach  ; 
420-r-,&=1200,  Ans. 

Ex.  34.  Since  he  is  to  cover  7%  of  the  cost  for  expenses, 
and  still  clear  12^%  on  the  cost,  the  sales,  without 
allowing  for  credit  or  bad  debts,  must  be  100%  + 
7%+12|%=119|%  of  the  cost.  But,  since  the 
sales  are  on  credit  of  6  mo.,  the  119.5%  is  the 
present  worth  of  the  sales;  and  119.5x1-03  = 
123.085%,  the  %  of  the  cost,  which  the  sales  would 
be,  without  allowing  for  bad  debts.  Now,  since  5  % 
of  the  sales  is  lost  by  bad  debts,  the  123.085%  of 
the  cost  is  95%  of  the  sales.  Hence 
123.085-i-.95=129.56+%,  what  the  sales  must  be 
to  cover  all  conditions  ;  and 
129.56—100=29.56+%,  Ans. 

Ex.  35.  28X20X10=5600  days'  work  by  1st; 

25X15X12=4500  «         "         2d; 

18X25X11=4950  «        "        3d; 

.       15X24X  8=2880  «         «        4th; 


17930       «         "  all. 

:^2686-00+j  lst  receives, 

=  2158.39+,  2d  « 

=  2374,24+,  3d  " 

=  1381. 37+,  4th  « 
Ex.36.  75XM«XTV^2284,^3. 

(419,  420) 


MISCELLANEOUS    EXAMPLES.  225 

Ex,  37.  $35.26+int.  for  101  da.=$36.15 
$48.65+  «  «  70  «  =  49.50 
$  6.48+  "  "  56  «  =  6.57 
$50.00+  «  "  30  "  =  50.38 

$142.60,  J.ws. 

Ex.  38.  Had  he  received  $.35  per  bushel  less  for  the  barley, 
the  whole  cost  would  have  been  $.35x56=$19.60 
less,  or  $63.10— $19.60=$43.50.     But  in  this  case, 
the  prices  of  barley  and  corn  would  be  equal.     Hence 
56+34=90  bu.,  whole  quantity; 
$43.50-r-90=$.48|,  price  of  corn; 
$.48|+$.35=$.83^,    «      «  barley. 

Ex.  39.  Since  $12950  is  the  proceeds  of  the  note  discounted 
for  6  mo.,  (allowing  3  days  grace),  $12950-*- 
.96441666=813427.81,-  face  of  the  note  re- 
ceived for  the  flour,  Dec.  1. 
$13427.81--  .85=$15797.42,  value,  Nov.  1; 
815797.42-5-1.30=  12151.86,  "    Oct.  1; 
812151.86-^1.25=     9721.49,  paid  for  flour; 
$12950— 89721.49=83228.51,  Ans. 

Ex.40.  $660-^120 =$5.50,  mean  price; 
5.50- 


J5.75UV 
15.0&I.V 

2 
1 

80 
40 

3 

120 

Ans.  80  bbl. 

From  the  conditions  <jf  the  question, 

6  da.  of  lst+10  d§.  of  2d=$38 ; 

3  da.  of  lst+  5  da.  of  2d=$19 ; 

and  15  da.  of  lst+25  da.  of  2d=$95. 

But  15  da.  of  lst+  8  da.  of  2d=$61 ; 

hence  17  da.  of  2d=$34  ; 

and    1  da.  of  2d=$2. 

(420) 


226 


MISCELLANEOUS    EXAMPLES. 


Now,  the  2d  mechanic  labored  8  da.-f-10  da.=18  da.  \ 
hence,  $2x18  =  $36,  2d  received; 

$61+$38— $36=$63,  1st     « 

Ex.  42.  $961.875-^-.15=z$6412.50,  net  value; 

$6412.50-^-$.125=r=51300  lb.,  net  weight ; 

180  lb.X-95=171  lb.,  net  weight  of  1  box ; 

51300-r-171=300  bags,  Ans. 
Ex.  43.  1/4900=70  lb.,  Ans. 

Ex.  44.  Had  he  worked  6  days  in  each  week,  he  would  have 
received  60X$2=$120,  and  paid  for  board  10X 
$4=$40,  thus  saving  $120— -$40=880  ;  hence 
$80— $72 =$8,  deduction  of  wages  for  idle  days; 
and  $8-i-$2=4  da.,  idle;  60— 4 =56  da.,  he  worked. 

Ex.  45.  $250Xl.806111=$451.528,  Ans. 

Ex.  46.  $120-^-100  =  $1.20,  price  per  gal.  of  wine  ; 
1 


831 
100 


Ans.  16|  gal. 


Ex.  47.  The  quantity  of  water  discharged  by  a  pipe  varies  aa 
the  area  of  a  section  of  the  pipe,  and  consequently 
as  the  square  of  the  diameter.  Hence 

1     f  3 

3   4  =1  :  3 

[2  UD 

(?)=2h.XfXiXf=4h.  30min.,  Ans. 

Ex.  48.  $4981.50-r-1.025=$f860  to  be  distributed ; 

$4860-v-$8100=.60=60%,  Ans. 

Ex.  49.  His  asking  price  was  125%  of  the  cost,  and  the  sell- 
ing price  was  1.25X.86=1.075=107^%  of  the 
cost ;  hence, 

(420,  421) 


MISCELLANEOUS    EXAMPLES.  227 

$170-f-.075=$2266.66f,  cost; 
$2266.66|+$170  =  $2436.66|,  sold  for; 
$2266.66f  Xl.25=$2833.33|,  asking  price. 

Ex.  50.  231  cu.  in.X2000=4620QO  cu.  in.,  solid  contents; 
t/462000=77.3+in.=6  ft.  5.3  in.+^rcs. 

Ex.  51.  21  yr. — 13    yr.=8    yr.,  1st  sum  at  interest ; 
21  yr.— 15    yr.=6    yr.,  2d     «  " 

21  yr.— 16^  yr.=4^  yr.,  3d     "  " 

Now,  $.06x8  =$.48,  int.  of  $1  of  the  first  sum, 
$.06X6  ==  .36,    «       "         "       second  " 
$.06X4^=  .27,    "       "         "       third    « 
And  $^3= present  worth  of  $1  of  the  first  sum; 
$T.10  =     "  "  "       "     second  " 

$T.iT=     "  "  "       "     third     « 

Hence,  the  $5000  must  be  divided  into  parts  pro* 
portional   to  T^F,  T|g  and  T|T;  or,  reducing  to  a 
common    denominator,    and   using   the  numerators 
(41 8 ?  HI),  the  proportional  terms  are 
17272, 18796,  20128.     Then 
17272+18796+20128  =  56196 
56196  :  17272=15000  :  (?)=$153tf.76+,  1st; 
56196  :  18796=$5000  :  (?)=$1672.36+,  2d; 
•    56196  :  20128=$5000  :  (?)=$1790.88— ,  3d. 

Ex.  52.  Let  A  B,  B  C,  C  D,  D  E,  E  F,    4^ 

and  F  Gr,  represent  the  distances        ^  « 
sailed  on  the  successive  days.     A 
G  will  be  the  distance  the  ship  de- 
parted from  A,  and  A  H+H  Q-,     H  F       G 
or  2  A  H,  will  be  equal  to  the  whole  distance  the 
ship  sailed.     Hence  A  GT2  =  118.794* =14112,  near- 
ly;JL4112--2=7056=A  H^ ; 
1/7056=84= A  H;  84x2=168  mi.,  Am. 
(421) 


228  MISCELLANEOUS    EXAMPLES. 

Ex.  53.    4  P.  x  110   =440  gal. ; 

$2.15  x  440=4946,  first  cost; 
$946  X  .24   =   227.04,  duty  ; 
5*7.60,  freight ; 

$1230.64,  whole  cost; 
$1980— $1230.64  =  $749.36,  gain; 
$749.36-7-$1230.64  — 60.89 -f  #,  Ans. 

Ex.  54.  We  form  equations,  as  in  Arbitration  of  Exchange, 
the  question  being  how  many  bushels  of  corn 
[(?)  corn]  are  equal  to  5  bbl.  of  flour. 
(?)  bu.  corn  =5  bbl.  flour; 
17  bu.  wheat=34.5  bu.  corn; 
59.5  bu.  oats=9  bu.  wheat; 
42  Ib.  flour   =6  bu.  oats; 
1  bbl.  flour   =196  Ib.  flour; 


(?)=42ff§,  Ans. 

Ex.  55.  100—8-^-92  %,  price  of  the  stock;  92  %  :  (0= 
10  %  :  7  %  (F)=Af3*=64.4  % }  100—64.4= 
35.6  °fo  discount,  Ans. 

Ex.  56.  £400 X  Y=*1777?>  old  par  value; 
$2000— $1777|=$222f,  premium; 
$222f-5-$17773=12£  %,  An*.  • 

Ex.  57.  21  mi. — 7  mi. =14  mi.,  B  gains  of  A  daily; 

84-s-14=6 ;  hence  A  and  B  are  together  every  6 
days.     7  mi.-{-14  mi. =21  mi.,  A  and  C  approach 
each  other  daily;  14  mi.-f-14  mL--28mi.,   B   and 
0  approach  each  other  daily;  84-j-21=4;  that  is, 
A  and  C  meet  every  4  days ; 
84-r-28=3  ;  that  is,  B  and  C  meet  every  8  days ; 
Now,  since  A  and  B  meet  every  6th  day,  A  and  0 

(421) 


MISCELLANEOUS   EXAMPLES.  222 

every  4th  day,  and  B  and  C  every  3d  day,  the  in- 
terval of  time  required  fcr  all  to  meet,  is  the  least 
common  multiple  of  6  da.,  4  da.  and  8  da.  =  12  da. 


Ex.  58. 


10463 

9436 
1027 

963 
64 


41852 

1348 
1027 


1 
4 


0463  _  1_ 
— 


321         The  approximating  fractions 
320     are  £,  &,  £,  f£,  and  J$f. 

Hence,  there  must  be  a  leap  year  once  in  four  years, 
7  times  in  28  years,  8  times  in  33  years,  31  times  in 
128  years,  or  163  times  in  673  years. 

Ex.  59.  $14071-5-10000=81.4071,  the  amount  of  $1  for  the 
time  he  owned  the  farm.  By  reference  to  the  table 
of  Compound  Interest,  we  find  that  the  time  in  which 
$1  at  5%  will  amount  to  $1.4071,  is  7  years. 
Now,  11  yr. — 7  yr.=4  yr.,  the  perpetuity  in  rever- 
sion when  he  purchased  it. 

$14071 X- 06=1844.26,  the  perpetuity  which  his 
money  would  purchase,  if  it  were  to  be  entered  upon 
immediately;  and  the  equivalent  perpetuity  in  rever- 
sion 4  years,  is  the  amount  of  $844.26  at  6%  com- 
pound interest,  for  4  years.  Hence, 
$844.06X1. 26247=$1065.85,  An*. 

Ex.  60.  $l-!-1.035=$.966=,  cash  value  of  $1; 
$1— 1966=1.034,  gain  on  $.966; 
$.034-=-1966=.035+,  Ans. 

Ex.  61.  The  several  investments  will  form  a  geometrical  se 
ries,  of  which  the  last  will  be  the  first  term  ;  the  last 
but  one  multiplied  by  1.05,  the  second  term-,  and  so 
on.  Hence  1.05=  ratio,  50 — 21=29=  No.  terms, 

(421,  422) 
20 


230  MISCELLANEOUS    EXAMPLES. 

and  $30000—  the  sum  of  the  series;  we  are  to  find 
the  first  term. 
Reversing  the  rule  under  T155  we  have 

$30000  X  (1.05—  1)        $1500 
-  -=—         —=$481.37,  Ans. 

1.0529—  1  3.116136 

Ex.  62.  The  shares  of  the  other  two  must  be  as  4  to  5  ;  hence 
4+5=9; 

$10000X|  =$4444|,  stare  Of  2d; 
$10000xf=$5555f,     «     "  3d. 

Ex.  63.  (?)hhd.=1500bbl.; 
50bbl.  =125  yd.; 
80yd.    =6  bales; 
13  bales=3|  hhd.; 

(?)=75T^  hhd.,  Ans. 

Ex.  64.  11|—  5     =6|  mi.,  the  7th  gains  of  the  1st,  daily  ; 
=5       "         "          «         "      2d      " 
—  3i.|  «        «          «         «      3d      a 

=3  «  «  "  «  4th  " 
1^—  9|  =1|  «  «  «  «  5th  « 
1|—  10|=1  «  «  «  "  6th  « 

120 
Hence,  The  7th  will  pass  the  1st  once  in  -  da.  f 

61 
«      ft      ((          2d     " 


120 
3(J     a     a    _    it 


5th  " 
6th  " 


a     a     12^     et 

120 


(422) 


MISCELLANEOUS    EXAMPLES.  281 

And  the  time  required  for  the  7th  to  pass  all  the 
others  at  the  same  place,  will  be  the  least  com- 
mon multiple  of  the  ahove  intervals,  which  is  found 
by  dividing  the  least  common  multiple  of  the  nume- 
rators by  the  greatest  common  divisor  of  the  denom- 
inators. Least  common  multiple  of  120  —  120  ; 
Greatest  common  divisor  of  6|,  5,  3{^,  3,  1|,  and  1 
=^2  ;  hence  120-^  =  1440  da.,  Ans. 

Ex.  65.  1  da.  =86400  sec. 

86400—20=86380,  No.  beate  1st  makes  in  1  da.; 
86400+15=88415,   «       "      2d      "       "      " 
||4og_4j2_Q  sec^  time  }n  which  lst  makes  1  beat; 


6400 
§4  1  5 


Now,  the  least  common  multiple  of  ||f§  sec.  and 
l^go  sec.  is  -L12^2.  sec.,  which  is  the  interval  of  time 
that  must  elapse  before  the  two  pendulums  will  beat 
in  unison.  But  in  this  time  the  1st  pendulum  will 
make  Ii28JL-=-||fj  =  2468  beats;  and  the  2d  pen- 


dulum will  make  !lf  £&-*-£  |  gf  =  24  69  beats.    There- 

fore, 2468  sec.=41  min.  8  sec.  P.  M.,  time  by  1st; 

2469  sec.  =41  min.  9  sec.  P.  M.,     "      «    2d. 

Ex.66.  (330—  l)x^     330—  1 

----  =  --  =343,518868244U  in.  ; 

3—1  6 

343,5188682441i-:-63360==541590730f!i£mi. 

Ex.  67.  105  %  :  85%=(?)  :  6  %  (?)==u^==71^  %. 

Ex.68.  A.      P. 

6  +  7  =$  .33,  1st  condition  ; 
10  +  8  =     .44,  2nd       « 
Istx5=30  +35  =$1.65 
=  1.32 
11  =$  .33 

(422) 


282  MISCELLANEOUS   EXAMPLES. 

$.33-^-11=    3  cents,  price  of  one  peach; 

$.03X7  =  1.21;  $.33— $.21  =  $.12,  price  of  6  apples; 

$.12-:-6=2  cents,  price  of  1  apple. 

Ex.  69.  If  A  has  $9 

B  will  have    5 
and  C  will  have  |  of  $5=    24 

$164 

- — =TIy53>  O's  part  of  the  whole  estate: 
164 

$3862.50-T-T'T51j=$29097.50,  Ans. 

Ex.  70.  C.      R. 

16  +20  =$30,  1st  condition ; 
24  +10  =  27,  2d         « 
1stX|=24+30=i45 
20=  18 

$18-;-20  =$.90,  price  per  bu.  for  rye; 
hence,  from  1st  or  2d,  $.75,         "         "          corn. 

Ex.  71.  For  1  ox  worth  $28, 
there  were  2  cows  "  34 
and  6  sheep  "  45 


$107 

=$196,  sum  paid  for  oxen 
749Xi%V=  238,  «          cows; 

749XT%\=  315,  sheep; 

hence,  $196-^$28  =  7  oxen  , 
238-^-  17  =14  cows; 
315-f-  7.50=42  sheep. 

Ex.  72.  $25000-7-.76=$32894.73+,"^w. 
Ex.  73.  The  difference  between  the  coat  and  $20,  and  the 
coat  and  $9,  is  $20 — $9=$11,  which  must  be  the 

(422,  423) 


MISCELLANEOUS    EXAMPLES.  2S8 

ratable  wages  for  20 — 12  —  8  weeks.     Hence 
$11-H8  =  $1|,  one  week's  wages  ; 
$lf  X20=$27.50,  wages  for  20  weeks ; 
and  $27.50— $20= $7.50>  value  of  coat. 

Ex.  11.  540  A.  36  P.  =  86436  sq.  rd.; 

1/86486  —  294:  rd.,  one  side  of  square  piece; 
86436-j-42=2058  sq.  rd.,  in  each  of  equal  square; 
1/2058  =  45. 3 -f-  rd.,  side  of  one  of  equal  squares. 

Ex.  75.  25  A.=4000  sq.  rd.;  1/4000  =  63.245+  rd.,  one 
side  of  the  square  field;  63.245  rd.X4=252.980 
rd.,  perimeter  of  the  square  field.  4000-^2=2000; 
l/2jUU=44.721  rd.,  width  of  the  rectangle;  44.721 
rd.X2:=89.442  rd.,  length  of  the  rectangle;  (44.721 
rd.,  +89.442  rd.)  X2=268.326  rd.,  perimeter  of 
the  rectangle;  268.326  rd.— 252.980  rd.  =  15.346 
rd.,  difference  of  the  perimeters ; 

1625X15.346  =  19.59+,  Ans. 

Ex.  76.  The  hour  and  minute  hands  start  together  at  12 
o'clock ;  at  1  o'clock  the  hour  hand  has  passed  over 
1  twelfth  of  the  dial,  while  the  minute  hand  has 
passed  over  the  whole  dial.  Therefore,  the  minute 
hand  gains  of  the  hour  hand  at  the  rate  of  {^  of  the 
dial  in  60  min.  But,  at  5  o'clock  the  minute  hand 
has  T\  of  the  dial  to  gain  of  the  hour  hand,  before 
passing  it.  Hence 

ji  :  T52=60  min.  :  (?)=27  min.  16T4T  sec. 
Ans  27  min.  16T4T  sec.  past  5  o'clock. 

Ex.  77.  It  is  evident  that,  to  increase  the  numher  in  both 
rank  and  file  by  1  man,  would  require  twice  the 
number  in  rank  and  file  at  first,  plus  1  (for  the  man 

(423) 


234  MISCELLANEOUS    EXAMPLES. 

at  the  corner).      And,  since  to  effect  this  requires 
284+25—309  men,  ££§=±=154  is  the  number  of 
men  in  rank  or  file  at  first. 
Hence  154*  +284=24000,^?. 

Ex.  78.  7  =first  proportional  term, 
9  =  second         "          " 
7  X|  =9£=  third 

25^  =  sum  of         "       terms. 

25^  :  7  =$3648  :  (?)  =$1008 
25^  :  9  =$3648  :  (?)  =$1296 
25|  :  9|=$3648  :  (?)  =$1344 

Ex.  79.  6  A.  3  R.  12  P.  =1092  P.     Then  by  691, 
13X21=273;  1092^-273=4;  1/4=2; 
13X2=26  rd.,  width  •  21x2=42  rd,  length  ; 
(26  rd.+42  rd.)X2=136  rd.  of  fence. 

Ex.  80.  A,  B,  0,  and  D,  do  J^-^jf^  Of  it  in  1  day; 
A,  B,  C,  and  E,   «  ^=7*1*1*       " 
A,B,D,andE,   «  T\,  =  T«l4fiF       " 

AP     T)     anrl   "R       «       I    --     5460  (t  « 

,  C/,  u,  ana  Jii,        yg  —  TosT4TJ 

BP     T)     pnrl    TT,       "       1    __     744 
,  U,  JJ,  ana  Jii  — 


__ 

14— 

Taking  the  sum  of  the  fractions,  we  shall  have  4 
days'  work  of  each  man.     Consequently 

A,  B,  C,  D,  and  E  do  y^ViV  of  Jt  in  4  daJs  ;  and 
A,  B,  C,  D,  and  E  «  ^4^  «  j  day> 
Hence,  414960^-3641  1=1  1^8^  da.  required  for 
all.  If,  now,  from  43T^VffU>  *^e  Part  °^  t^ie  wor^ 
which  all  could  do  in  1  day,  we  take,  successively. 
the  parts  which  B,  C,  D,  and  E  ;  A,  C,  D,  and  E  ; 
A,  B,  D,  and  E  ;  etc.,  can  do  in  one  day,  we  shall 
have  the  parts  which  A,  B,  C,  D,  and  E  can  do, 
separately,  in  1  day,  thus  : 

(423^ 


MISCELLANEOUS    EXAMPLES.  235 


»  A  Cim  do  i 

36411  6916    __    8747        T*  U  (I 

JttGTj  -  TZT3T40  --  4r4"" 


Therefore,414960-v-4491  =  92|f|f  da.  req'rd  for  A  ; 
414960--8747=  47|||f  "          B; 

414960^-1931=214i?|f          «          0  5 
414960-^-4571=  90|f^  "          D; 

414960-T-6771—  61^9T  "  E. 

Hence  B  will  do  the  work  in  the  shortest  time. 

.Ex.81  The  first  has  1    share; 

the  second  has  1   Xli=H       " 
the  third  has 


All  have  4|       « 

4|  :  1  =$500  :  (?)=-4105T\,  1st  has  ; 
4|  :  U=$500  :  (?)=$157j|,  2d    « 
4|  :  2i=$ 


Ex.  82.  $2500X5          =$12500  for  1  mo.  ; 
$5500X7          =  38500  "      « 


B's  investments=$51000  «       a 
S1333^---51000=$T|3,B's  (and  also  A's)  gain  per 
month  on  each  $1  of  capital  invested. 

$5000 Xyfs  X4=$522if|,  A>s  gain  during  the  first 
4  months ; 

$1066|— 8522i||=$543i|i,  A's  gain  during  the 
last  8  months ; 

$T|3 x8  =  $T%,  gain  on  $1  in  8  mo.;         

$543if!-f-T%23=$2600,  A's  capital  for  last  8  mo.; 
S5000— $2600=$2400,  Ans. 

(423)  f     ' 


236  MISCELLANEOUS   EXAMPLES. 

Ex.  83,  There  will  be  4  intervals  of  time  each  ^  year,  and 
the  rate  5  %  ;  hence 
$12750-=-1.215506=$10489.459-,  Ans. 

Ex.  84.  The  proceeds  of  $1  for  93  da.  are  $.9819 £ ;  and  since 
he  sells  at  120  %  of  the  purchase  price,  he  realizes 
120  %X.9819i=ll7.83  %  in  ready  money.  De- 
ducting 5  %+2  %=7  %  for  expenses,  we  have 
117.83  %—  1  $,=110.83  %,  his  rate  of  net  receipts 
on  each  investment.  Hence 

$1500      X  1.1083=81662.45,  net  proceeds  1st  sale; 
$1662.45x1-1083=  1842.53+,  «      "        2d    " 
$1842.53X1.1083=2042.07+,"      "        3d    " 
$2042.07X1.1083=  2263.23+,  «      «        4th  « 
$2263.23— $1500  =     763.23,  whole  gain,  Ans. 

Ex.  85.  $.80X1.20=1.96,  what  it  must  sell  for;  and  since 
the  selling  price  is  100  %— 10  %=90  %  of  the 
asking  price,  we  have  |.96-*-..90==fl.06|,  Ans. 

Ex.  86.  25X20=500  yd. 

$4JX500— $2187.50,  purchase  price; 
$4|X500=:  2312.50,  selling         " 
$2312.50-f-1.02  =$2267.15,  cash  value  of  sale ; 
$2187.50-i-1.045=  2093.30,  «          purchase. 

$173.85,  Ans. 

Ex.  87.  $1200^-200=6,  number  of  payments ; 
(1.086— 1)X200 
=$1437. 185,  amount  of  the  annuity 

.08 

at  the  maturity  of  the  last  installment.     And  sin  ce  this 
sum  will  be  at  compound  interest  for  10 — 6=4  years, 
$1467.185xl.360489=$1996.07+,  Ans. 
(423,  424) 


MISCELLANEOUS   EXAMPLES.  2f,7 

Ex.  88.  The  amount  of  an  annuity  of  $3000  in  arrears  for 

(1.06 10— 1)X$3000 

10  years,  is =$39542.40. 

.06 

The  present  worth  of  this  sum,  due  10  years  hence, 
is  $39542,40-1-1.790848=822080.28,  the  value  of 
the  bequest  to  the  eldest  son. 

The  present  value  of  the  same  sum  due  20  years 
hence  is  ^39542.40--3.207136=;$12329.51,  the  val- 
ue of  the  bequest  to  the  second  son. 

The  worth  of  the  perpetuity  when  entered  upon  by 
the  institution,  was  $3000-f- .06 =$50000. 

But,  as  the  perpetuity  is  not  to  be  entered  upon  till 
20  years  after  the  man's  decease,  its  present  value  is 
$50000-i-3.207136=$15590.23. 

Ex.  89.  It  requires  5-|-l=6  days  for  the  horse  team  to  per- 
form the  trip  and  rest  a  day,  7+1=8  days  for  the 
mule  team,  and  11-|-1=12  days  for  the  ox  team  to 
do  the  same.  Now,  the  least  common  multiple  of  6 
da.,  8  da.  and  t2  da  is  24  da.  Therefore  the  three 
teams  will  rest  together  on  the  24th  day,  and  conse- 
quently 24 — 1=23  da.  must  elapse  before  this  day 
comes. 

Ex.  90.  The  four  payments  must  be  treated  according  to  the 
U.  S.  rule  for  Partial  Payments ;  that  is,  the  pay- 
ment each  year  will  cancel  the  interest  which  has 
accrued  on  the  principal  for  that  year,  together  with 
a  certain  portion,  or  installment,  of  the  principal  it- 
self. Now,  the  principal  for  the  second  year  will  be 
less  than  the  principal  for  the  first  year,  by  the  value 
of  the  first  installment;  and  consequently  the  inter- 
est to  be  canceled  by  the  second  payment  will  be 
(424) 


238  MISCELLANEOUS   EXAMPLES. 

less  than  the  interest  canceled  by  the  first  payment, 
by  one  year's,  interest,  or  6  %,  of  the  first  install- 
ment. Hence  the  second  installment  will  exceed  the 
first,  by  the  same  sum  ;  that  is,  the  second  install- 
ment will  be  1.06  times  the  first.  For  a  similar 
reason,  the  third  will  be  1.06  times  the  second  ;  and 
so  on.  Therefore,  these  installments  of  the  princi- 
pal form  a  geometrical  series,  of  which  the  1st  in- 
stallment =±=  first  term,  4=  No.  terms,  and  $4500= 
sum  of  the  series.  We  may  find  the  first  term  by 
reversing  the  rule  under  7155  thus: 

(1.06—  1)X  $4500 

--  =$1028.67+=  1st  term,  or  that 
1.06  4—  1 

part  of  the  principal  canceled  by  the  first  payment. 
But  the  payment  must  also  cancel  the  interest  of  the 
principal  for  that  year,  which  is  $4500X-06=$270; 
hence  $1028.67+$270=$1298.67,  Am. 

Ex.  91.  Of  the  payments  coming  due  after  Jan.  1,  1864, 
one  is  to  be  discounted  for  2  half  years  at  6  %  per 
annum,  or  for  2  intervals  at  3  %  ;  and  the  other  for 
4  intervals  at  3$.  Hence 

$1050,  4th  payment; 

$1050-7-1.0609=$  989.725+,present  val.  5thpa/t. 
$1050-4-1.125509=     932.911  +,         "        6th    " 


$2972.636+, 

Ex.  92.  2X3=6  ;  864-4-6=144  ;     T/144=12  ; 
3x12=36,  No.  of  rows; 
2x12=24       "       trees  in  each  row. 
And  since  the  spaces  between  the  trees  are  1  less, 
each  way,  than  the  number  of  trees, 

(424) 


MISCELLANEOUS    EXAMPLES.  289 

(36  —  1)X7=245  yd.,  length  of  the  orchard; 
(24-  -1  )  X  7  =  161  yd.,  breadth      « 
245X161=39445  sq.  yd.,  Ans. 

Ex.  93.  $2500X1.06=12650,  first  cost; 

$2650x1-225043=  $3246.36,  amount  for  3  yr.; 
$3246.36X1.0175=  3303.17,         "       3  yr.  3  mo, 
which  is  to  be  considered  as  the  whole  cost  of  the 
investment. 

Now,  there  were  received  6  dividends  of  $100  each, 
received  2  yr.  9  mo.,  2  yr.  3  mo.,  1  yr.  9  mo.,  1  yr. 
3  mo.,  9  mo.,  and  3  mo.,  respectively,  before  the  sale 
of  the  stock.  Hence,  reckoning  compound  interest, 
Amt.  of  $100  for  2  yr.  9  mo.  =$120.50 
"  .«  2  "  3  «  =  116.49 

«  «          1  «    9   "  =  112.62 

«  "          1  "    3   "  =  108.87 

«  «  9   «  =  105.25 

«  «  3  "  =  101.75 

Value  of  dividends,  $  665.48 
Sale  of  stock,  $2500x1.11=  2775.00 

Whole  sum  realized,  $3440.48 
$3440.48—  $3303.17=4137.31,  Ans. 

Ex.  94.  Firstj  the  number  in  the  smallest  company  is  4,  and 
in  the  largest  64,  and  the  number  in  each  company 
is  double  the  number  in  the  preceding  company. 
Hence  the  companies  are,  respectively,  4,  8,  16,  32, 
64.  There  are  therefore  5  companies,  and  the  total 


number  of  men  is  -  =124. 

2—1 
Second^  The  smallest  company  received  $.50  each 

C424) 


24:0  MENSURATION. 

per  day,  tin  largest  $1.50  each,  and  the  common 
difference  of  the  rates  of  daily  wages  is  $.25  Hence 
the  daily  rates  in  the  respective  companies  are  $.50, 
$.75,  $1.00,  $1.25,  $1.50. 

Now,  if  we  multiply  these  rates  of  daily  wages,  each 
by  the  number  of  men  in  the  respective  companies, 
and  add  the  products,  we  shall  obtain  the  whole  sum 
paid  per  day,  thus  : 

$  .50,     $  .75,     $  1.00,     8  1.25,     $  1.50 
4  8  16  32  64 

$2.00  +$6.00  +$16.00  +$40.00  +$96.00  =$160 

paid  per  day  \ 

$160X6=?960,  weekly  payment. 


MENSUKATION. 

(731 5  page  425.) 

Ex    (.     (24+15)  X2=78  ft.,  perimeter  of  the  room; 
78X83       =663  sq.  ft.  in  the  walls; 
24X15X2=720  «    "   in  floor  and  ceiling ; 

1383  sq.  ft.  =  153|  sq.  yd.,  Arts. 

Ex.  U.     Let  A  B  0  D  E  F  represent  the  farm.  B u 

If  we  extend  the  line  D  E  to  Q-,  the  G 

land  will  be  divided  into  two  rectangles, 

G  B  C  D,  and  A  O  E  F.     Since  B  0=  A  _ 

25.14  ch.,  and  C  D  =12.08  ch.,  we  have  25.14X 

12.08=303.6912  sq.  ch.,  area  of  G  B  C  D.     Again, 

since  E  F  =26.12  ch.,  and  F  A  =16.84  ch.,  26.12 

X  16.84=439.8608  sq.  ch.,  area  of  A  a  E  F.     Then 

(424,  425) 


LINES   AND   SUPERFICIES. 


2-n 


439.8608+303.6912=743.552  sq.  ch.=74.3552  A. 
=74  A  56.8+  P.,  Ans. 

Ex.  3.     240-f-20  =  12  rd.,  Ans. 

Ex.  4.     70  A.=11200  sq.  rd.  ;  11200-5-120=93^  rd.,  Am. 

Ex.  5.     40  A.  =400  sq.  ch.  ;  400—8=50  ch.,  Ans. 


72 


(732,  page  426.) 
.fix.  1.     28  ft.  9  in.=28|  ft.; 

36X28|  =  1035  sq.  ft.  =3  sq.  rd.  218|  sq.  ft.,  Ans. 

Ex.2.    1/202—  122=16; 
72X16=1152  P. 

=7  A.  32  P.,  Ans. 

Ex.  3.     14  A?72  P.=2312  P.     Now,  by  691,  1X2=2, 

product  of  the  proportional  terms;  2312-1-2=1156; 
1/1156=34;  34X1=34,  shorter  side;  34x2=68, 
longer  side;  and  68x30=2040  P.=12  A.  120  P. 

(733,  page  426.)  * 

Ex.  1.     12^+8jbJL<r=78.38  ch.,  half  sum  of  parallel  sides  ; 
78.38X30.2§=2370.995  sq.  ch.=237  A.  15.92  P. 

Ex.  2.     ^4^=12^  in.,  average  width; 

(12£Xi2)-j-12=12£  sq.'ft.,  Ans. 

iUL£L4=45  rd.  ; 

45+24=1080  P.=6  A.  120  P.,  Ans. 


.  3. 


54 


(734,'page  427.) 
Ex.  1.     126X\4=1^12  sq.  ft.=168  sq.  yd.,  Ans. 

Ex.  2.  The  two  gable  ends  to- 
gether make  a  parallel- 
ogram, whose  length  is 


17 


1425-427) 


242 


MENSURATION. 


34  ft.,  and  perpendicular  8  ft.     Hence 
34  X  8=272  sq.  ft.,  Ans. 

Ex.  3.     48-r-12=4  ft.  =  one-half  the  perpendicular; 
4  ft.  X  2 =8  ft.,  Ans. 

Ex.  4.     108--9  =  12  ft.;  12  ft.>(2=24  ft.,  Ans. 
Ex.  5.     -*-£^=135  sq.  rd.,  Ans. 

Ex.6.  1/182— 92=15.588 
+  ft.=BD;  15.588 
X  V8=140.29+sq.  in. 


Ex.1. 
Ex.2. 
Ex.  3. 


Ex.4. 


*    (735,  page  428.) 

8  ft.X3.14159=25.13272  ft.=25  fy  1.59+  in. 
49.52X.31831=15.762+  rods,  Ans. 

4  ft.  8£  in.=56.5  in.;  56.5x2=113  in.,  diameter; 
113x3.14159=355 —  in.,  circumference; 
18° =3^  =  2*0  of  a  circumference; 
355  in.X2V=17-75  in— 1  ft-  5-75  in^  Ans- 
66  ch.=264  rd. ; 

264  X.  31831=84.03384  rd.,.diameter  of  garden; 
66X.31831=21.00846   "          "        "pond; 


2)  68.02558  rd.,  twice  the  width  of  ring ; 

31.51+  rd.,  Ant. 
16.5X-31831=5.25+  ft.=5  ft.  3+  in.,. 


Ex.5. 

(736,  page  428.) 
Ex.  1.    ^21x^10^40115,  Ans. 
Ex.2.    25*  X-  7854=490.875  sq.  in.,  Ans. 

Ex.  3.     6  ft.  10  in.=82  in. ; 

82*X-07958=535.1+sq.  in.=3  sq.  ft.  103.1  sq.  in. 

(427,  428) 


SOLIDS.  2-13 

Ex.  4.  6.44598-r-.07958=81  ch.,  square  of  circumference  ; 
l/8l=9  ch.=9  ch.=36  rd.,  ^ns. 

Ex.  5.     82X-7854=50,2656  sq.  rd.,  area  of  circle ; 
50.2656-=-4=12.5664  sq.  rd.,  Ans. 

(744,  page  429.) 
Ex  1.     16 3 =4096  cu.  ft.,  Ans. 
Ex.  2.     15X3X  Ji=41i  cu.  ft.,  Ans. 
Ex.  3.     l/ia— .5a=V776"=  .866  ft.+,  altitude  of  triangle ; 

.866X.5=-433  sq.  ft.,  surface  of  the  end; 

24x.433=10.39+cu.  ft.,  Ans. 

Ex.  4.    .?854X92X9^X1728=1044343.2384  cu.  in. ; 
1044343.2384-f-231=4520.96+gal.= 
71  hhd.  47.96+gal.,  Ans. 

Ex.  5.    Let  A  B   C  D  be  tlie  end  of  tlie 
log  when  hewn  square. 
B~C2+BA2  =  2  BO^AC^^ 
Hence,  BG2=rr50  sq.  ft.,  the  area 
the  square  end. 
.7854X10* X20X12=18849.6  cu.in.  in  whole  log; 
50  sq.  ft.  X  20x12=12000         "      «  sq.timber; 

6819,6  cu.  in.=3  cu.  ft. 
1665.6  cu.  in.,  Ans. 

Ex.  6.  1  h.  20  min.=4800  sec.  Hence  the  quantity  of 
water  discharged  will  be  equal  to  the  contents  of  a 
tube  2  in.  in  diameter,  and  12  in.  X  4800 =57600  in, 
in  length.;  .7854 X22X 57600=180956.16  cu.  in. ; 
180956.16  cu.  in.-f-231=783.36+gal.,  Ans. 

Ex.  7  29— 23=6;f  of  6=4;  23 +4=27  in.,  mean  diameter. 
,7854x27^X36 

=.0034X27*X36=89.2296  gal. 

231 

(428-430) 


244  MENSURATION. 

Ex.  8.     28—25=3;  T60  of  3=1.8;  25+1.8=36.8  in.,  mean 

diameter.     And  since  .7854--231=.0034, 
.    .0034X26.8*  X35=85.47056  gal.,  Ans. 

(745,  page  430.) 

Ex.  1.    173X12=3468  cu.  ft.,  solid  contents. 

From  the  foot  of  the  perpendicular  to  the  middle  of 
one  side  of  Che  base  is  17-4-2=8.5  ft.     Hence, 
1/36*  +8.52  =36.99+ft., the  slant  height;  36.99 X 
8.5x4=1257.66+sq.  ft.,  lateral  surface. 

Ex.  2.    5i|  ft.=5.916666+  ft. ; 

5.9166662X-  07958=2.78584+  sq.  ft.,  area  of  base; 
2.78584X|=4.64306+  cu.  ft.,  Ans. 

Ex.  3.     From  the  solution  of  Ex.  5,  74:45  we  learn  that 
the  square  of  the  diameter  of  a  circle  is  equal  to 
twice  the  area  of  the  square  that  may  be  drawn  with- 
in it.     Hence, 
72 

— =24.5  sq.  ft.,  area  of  the  base  of  the  pyramid ; 

2 

24;5X1/=24.5X  6=147  cu.  ft.,  Ans. 
Ex.  4.    £&!^-X4==1500  sq.  ft.  in  the  convex  surface ; 
302  =  900  "     "      " 


2400  sq.  ft.  in  entire  surface. 

Now,  since  the  slant  height  is  25  ft.,  and  a  line  from 
the  middle  of  one  side  of  the  base  to  the  foot  of  the 
perpendicular  is  15  ft.,  we  have 
1/252— 152=20  feet,  the  altitude; 
and  302X23°:=:6000  cu-  ft->  solid  contents. 

Ex.  5  30X  V8=270  sq-  in—1  sq.  ft-  126  sq-  in->  convex 
surface;  30X-31831=9.5493  in.,  diameter  of  the 
base;  9.5493  in. -=-2 =4. 7 7465  in.,  radius.  Now, 

(430,  431) 


SOLIDS.  245 

since  the  slant  height  of  the  cone  and  the  radius  of 
the  base  form  the  hypotenuse  and  base  of  a  right- 
angled  triangle,  of  which  the  other  side  is  the  altitude 
of  the  cone, 


.774652=17.3580+  in.,  altitude  ; 
302X-  07958=71.622  sq.  in.,  area  of  base; 
71-  fi?3g  i?-35  8;—  [243.21+  cu.  in.,  solidity. 

(746,  page  431.) 

Ex.  1.    42X3.1416=50.2656  sq.  ft.,  Ans. 
Ex.  2.     8.53X.5236=321.55+cu.  in.,  Ans. 

123x.5236 
Ex.  3.     --  =452.39-|-cu.  in.,  Ans. 


Ex.  4.  Since  the  corners  of  the  inclosed 
cube  will  touch  the  surface  of  the 
sphere,  the  diagonal  of  the  cube, 
E  C,  will  be  the  diameter  of  the 
sphere,  or  12  ft.  Now,  remembering  that  A  E,  A 
B,  and  B  C,  are  all  equal,  being  each  a  side  of  the 
cube,  we  have  (A  E)2+(A  C)2=(E  C)2.  But  (A 
C)2=(AB)2+(BC)2,  or  2(AB)2,  or  2(A  E)2. 
Hence,  3 (A  E)2=(E_C)2=122=144,  and  144-- 
3=48=(A  E)2  ;  1/48=6.928+  ft.=  A  E.,  Ans. 

Ex.  5.     65.45-4-;5236=125,  cube  of  diameter; 
^125=5  in.,  Ans. 

Kx,  6.  Keversing  the  rule  for  finding  the  surface,  78.54-r- 
3.1416=25,  the  square  of  the  diameter;  and  1/25 
=5  in.,  the  diameter.  Then,  by  the  rule -for  finding 
the  solid  contents,  53X.5236=65.45  cu.  in.,  the 
capacity. 

(431) 


24:6  MENSURATION. 

Ex.  7.     14 3X- 5236  =i  1436.7584  cu.  in.  in  larger  globs, 
12 3 x. 5236=  904.7808  «     "    «   smaller  « 


531.9776  cu.  in., 

(747,  page  432.) 

ffix.  1.     1  gal.=231  cu.  in.     Hence  231-^-2=115.5  cu.  in., 
the  contents  of  the  lead;  $.15x115.5  =  117.325. 

Ex.  2.     6  gal.  3  qt.  1  pt.=6|  gal. ";  8|  gal.— 6|   gal.=l| 
gal.,  the  solidity  of  the  anvil.     Hence, 
231  cu.  in.Xlf =317f  cu.  in.,  Ans. 

Ex.  3.  83=512      cu.  in.,  contents  of  box; 

8j  qt.=j|  gal.=216T9g  cu.  in.,  deficiency; 

295T7g  cu.  in.,  Ans. 

(748,  page  432.) 

Ex.  1.     The  diameters  will  be  to  each  other  as  the  cube  roots 
of  64  and  512,  as  4  to  8,  or  as  1  to  2. 

Ex.  2.     Since  the  blocks  are  to  each  other  as  105_to  2835, 
or  as  1  to  27,  the  lengths  will  be  as  ty  \  to  "^27, 
or  as  1  to  3.     Hence  7  in.  X  3=21  in.,  Ans. 
Or,  105  :  2835=7  3  :  (?)=9261 
1^9261=21  in.,  Ans. 

Ex.  3.     iTT  :  lTir=:2  ft.  :  ('0=2  ft.  X  1.4422=2.8844  ft. 
=2  ft.  10.6+  in.,  Ans. 

Ex.  4.     Since  the  contents  of  the  required  cellar  will  be  18 
-4-6=3  times  the  contents  of  the  given  cellar,  its  seve- 
ral dimensions  will  be  ty  3  =1.4422  times  the  given 
dimensions.     Hence 
14  ft. X  1.4422=20.1908  ft.,  length; 
12  ft. X  1.4422=17.3064  ft.,  width; 
6  ft.Xl.4422=  8.6532  ft.,  depth. 

(431,  432) 


SOLIDS.  247 

« 

Ex.  5.     9000—217=41.474+,  ratio  of  the  contents  ; 


=3  .46-}-,  ratio  of  respective  dimensions  ; 
20  in.XB.46=69.2+  in.,  length; 
15  in.x3.46=51.9+  in.,  breadth; 
8  in.X3.46=27.6+  in.,  thickness. 

Ex.  6.  A  will  take  off  a  pyramid  containing  4  tons  ;  A  and 
B,  a  pyramid  containing  8  tons  ;  and  A,  B  and  C,  a 
pyramid  containing  12  tons.  And,  since  these  pyra- 
mids are  similar  figures,  they  will  be  to  each  other  as 
the  cubes  of  their  altitudes.  Hence,  comparing  the 
whole  pyramid  with  each  pyramid  taken  off, 

Tons.     ft. 

.16  :    4=163  :  (?)=1024,  cube  of  height  taken  by^  ; 
16  :    8=163  :  (?)=2048,        "  "    A  and  B; 

16  :  12=163  :  (?)=3072,         "  " 

^1024=10.079  ft.,  taken  off  by  A; 

^2048=12.699"     "        "       AandB; 
.  ^3072=14.537  "     "        «       A,  B  and  0 

12.699—10.079=2.620  ft.,  taken  off  by  B  ; 

14.537__f2.699=1.838  «  "          C; 

16.000—14.537=1.463  «  "          P 

(432) 


A 


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